SEP 1 5 1966

^ecltnlccii rlote 92o. 342

HYDROMAGNETIC WAVE PROPAGATION NEARIc/s

IN THE UPPER ATMOSPHERE AND THE PROPERTIES

AND INTERPRETATION OF Pc 1 MICROPULSATIONS

JOHN A. DAWSON

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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* NBS Croup, Joint Institute for Laboratory Astrophysics at the University of Colorado. ** Located at Boulder, Colorado.
NATIONAL BUREAU OF STANDARDS

technical <v{©te 342

ISSUED June 30, 1966

HYDROMAGNETIC WAVE PROPAGATION NEAR 1 c/s IN THE UPPER ATMOSPHERE AND THE PROPERTIES AND

INTERPRETATION OF Pc 1 MICROPULSATIONS

John A. Dawson

Institute for Telecommunication Sciences and Aeronomy*

Environmental Science Services Administration

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

* Formerly the Central Radio Propagation Laboratory of the National Bureau of Standards.

CRPL was transferred to ESSA in October 1965, and will temporarily continue to use NBS

publication series until inauguration of their ESSA counterparts.

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington, D.C. 20402

Price: 40$

CONTENTS

PART I Page

Hydromagnetic Wave Propagation near 1 c/s in the Upper Magnetosphere

1. Introduction 1

2. Hydromagnetic Propagation in a Uniform Cold Plasma • • • • • 3

3. The Effect of Ion-Electron Collisions 12

4. Justification of Assumptions •••• 15

Figure 1 17

Figure 2 19

Table 1 20

Table 2 22

Table 3 25

5. Propagation in a Multicomponent Plasma 26

Figure 3

'. 33

Table 4 35

Figure 4 • 36

Table 5 38

Figure 5 . . . . 39

PART II

The Properties and Interpretation of Pc 1 Micropulsations

1. Introduction •••• • •• 41

Figure 6 ....... 42

Figure ? 43

2. Properties of Pc l's 46

3. Theoretical Interpretation of Pc l's 50

in

Table 6 51

Figure 8 53

Figure 9 • • • •••••• 56

Acknowledgments 62

References ••••••• 63

HYDROMAGNETIC WAVE PROPAGATION NEAR 1 c|s IN THE UPPER ATMOSPHERE

AND THE PROPERTIES AND INTERPRETATION OF Pc 1 MICROPULSATIONS

by

John A. Dawson

PART I

Hydromagnetic Wave Propagation near 1 c/s in the Upper Magnetosphere

Dispersion and polarization relations are developed

from basic considerations for a hydromagnetic wave

at about 1 c/s propagating in a cold uniform plasma.

The treatment is then extended to include the effects of ion-electron collisions. It is shown that

the approximations used are valid in the earth's

magnetosphere for heights between 10,000 and 50,000

km (2.6 to 8.8 earth radii). At lower elevations

the plasma must be considered as nonuniform. Above

50,000 km the small amplitude assumption is invalid.

Finally the theory is extended to include the effects of a multicomponent plasma.

1. Introduction

The subject of hydromagnetic wave propagation in all its complexity

has been treated extensively in the literature. The purpose of this

paper is to develop those aspects of the subject which are particularly

apropos to the propagation of hydromagnetic waves near 1 c/s in the

upper magnetosphere. It was considered worthwhile to develop the theory

from elementary considerations, to keep in mind the various assumptions

made in order to keep the problem tractable, and to justify their application to the magnetosphere.

The assumption of a uniform plasma is commonly made, so that the

solutions will have the form of plane harmonic waves. However, as will

be shown later, such an assumption is not really valid at heights below

10,000 km. In this region the plasma density changes significantly within

one wavelength.

Another assumption often made is that of a cold plasma, i. e.,

all collisions and hence, all pressure terms and momentum exchange be-
tween classes of particles are neglected. This is valid as long as

the collision frequencies are less than the wave frequency, which assumption can safely be made for the frequency range of interest for

heights above 3000 km, but lower than this, ion-electron collisions

become important. For the magnetosphere this assumption is less restrictive than that of a uniform plasma.

The small amplitude assumption is usually made, which allows the

equations to be linearized. This condition is invalid for large pulsations as the limits of the magnetosphere are approached. However, micropulsations in the 1 c/s range seldom exceed a few gamma, and the

assumption can safely be justified for heights lower than 50,000 km.

The plasma will be considered to consist only of electrons and

ions. Effects of neutral particles may be ignored as long as the wave

frequency is greater than the ion-neutral collision frequency. This

condition is met by staying above 300 km. Since the atmosphere consists

largely of hydrogen above 3000 km, only one type of ion need be considered. The plasma will be assumed to be neutral and the ions to be singly

ionized, i. e., n = n.

.

' ' e i

Thus, analysis of small amplitude plane waves in a cold plasma

is sufficient to explain hydromagnetic wave propagation in those regions

with heights between 10,000 and 50,000 km, but more elaborate treatments

will be required elsewhere. It is this region which will be considered

in this paper.

Mks rationalized units will be used in the following derivations.

Useful references in the field are Alfven and Falthammar [1963],

Spitzer [1962], Stix [1962], and Denise and Delcroix [1963].
2. Hydromagnetic Propagation in a Uniform Cold Plasma

The purpose of this section will be to derive a dispersion relation, suitable for all directions of propagation, for hydromagnetic

waves at frequencies around 1 cA, The polarization properties of these

waves will also be discussed.

We shall start with Maxwell's equations,

V X B = mJ + m«~ (1)

VXE = -f (2)

and define the conductivity tensor by

J = a . E (3)

The important distinction between a hydromagnetic wave and an electromagnetic wave is that the first considers the material current

important while the latter emphasizes the displacement current. As

will be shown in section four, the displacement current can be neglected

in the magnetosphere for ultra-low frequencies. However, under the

ionosphere the displacement current is important and propagation becomes

electromagnetic

Let B be composed of a steady component, B , which is derivable

from a multipole potential expansion, and a time varying component,

-* .- . .-» . b, which is small, i. e., |b| « |B |. Neglect the displacement cur-

~* ~* ~* —icot

rent. Let b, J, and E vary as e , which states that the various

plasma parameters do not change significantly during one period. Since

dB ~* o V X B = -rr— = 0, Maxwell* s equations become
V X b = m J (4)

and

V X E = iwb (5)

Take the curl of (5), and substitute from (4) and (3) to obtain the

wave equation

V X (VIE) = iWf*(o . E) (6)

So far nothing has been stated concerning the structure of the plasma,

nor the form of the conductivity tensor, other than that it is of second

order.

The conductivity tensor for a cold plasma can be obtained by considering the equations of motion for electrons and ions, and the definition of current density.

dv

medT = - iwm

e^e = -^

+ V *o }

TM

dv. _^ m

± 3— = - iam^ = e(E + v

±

X B

Q

) (8)

J = n

ee(v. - v

e

) (9)

Let B lie along the z-axis, so that (7) becomes in rectangular coordinates

- iam v = - e(E + v B ) (10)

e ex v x ey o

v '

- iam v = - e(E - v B ) (11)

e ey v y ex o

y K J

- iam v = - eE (12)

e ez z

v '

Solving for v , v , and v gives
2

iam eE + e B E

v =

e_^c o^

(13) ex 2D

2 2 2

K±^ J e B - a) m

o e

2 - e B E + iam eE

o x e y

ey 2„2 2 2 J e B - to m

o e

ieE

z V =

ez am

e

Similarly,

2 - iom.eE + e B E

(14)

(15)

v = L_JS o_z (16) ix 2^2 2 2 u° ; e B - o> m. o 1

- e B E - iom.eE v = 2_* ^_Z v ( 1? ) iy 2n2 2 2 K±u

J e B - a) m. o i

ieE

v, = ^ (IS) iz am.

v ' i

Make the following approximations

2

0) « 0) , a>. « a> . u « 0) a>. e' l e' e i

eB eB

where a) = and a), = are the gyrofrequencies. The second inequality

e i

is equivalent to m « m. . Then from (9)

2 2 2 in m.e o)E n m.ea) E

x ~ 2_2 2 2

T

_ , 2n2 2 2, Kxy)

e B - m.o) B (e B - m.w ) o 1 1

2 2 2 n m.ea) E in m.e a)E

t e_i x e l y_ , 2

* V ~ d t 2v

2 2 ^ 2D2 2 2 ^u;

* B (e B - m.o) ) e B - m.o)

O

v O 1 O 1
J

z *

in e lS e z m co e

Thus, it can be seen that the conductivity is given by

a =

*°1 °2 °

"°2 °1 °

a J

o

where

in e

e m oj e

2

in m.e co e 1

2D

2

e B -

2 2

m.co l

2 2 n m, eco e i

B (e B - m.co ) o

v o l

'

(21)

(22)

(23)

(24)

(25)

a, is often called the Pedersen conductivity, while o

? is the Hall conductivity.

-* To obtain the dispersion relation for a plane wave, allow E to

-» -> ik r

~* vary spatially as e • Thus, V X becomes ik X and the wave equation

(6) becomes

.= -*. - k X (k X E) = icoM(o . E) (26)

Without loss of generality, let k lie in the z-x plane at an angle 9

with respect to B , so that

(27)

k = k Xsin 6

k =

y

k = k

z cos e
Writing (26) out in rectangular coordinates using (22) and (27) gives

2_ 2 2

k E cos 6 - k E sin 6 cos 9 = iuWo.E + o E ) x z

v 1 x 2 y

k^ = ia)M(- o

2E

x

+ o^ )

- k 12. sin 6 cos + kTl sin 6 = ia)yua E

(28)

X Z

' o z

In order for (28) to have nontrivial solutions, the following determinant must be zero.

2 2 2

k cos 9 - iw^o., - iu)MOp - k sin 9 cos 9

iiejua.

- k sin 9 cos 9

k - iaj/uLO",

2 2

k sin 9 - ioo/AO

= (29)

This yields a quadratic in k

(a cos

2

9 + a.. sin

2

9)k

L. - lu>ii[o a (1 + cos2*2222

9) + (a.. + o„)sin 9]k

- u » a (o~ + o ) =

o 1 2

which produces the following dispersion relation

iM

o a.. (1 + cos 9)

° X - + sin2

9

2 _, 2

°l

+ °2

(30)

[22 2^ / . <* /4oo

a

2 cos 9 / o^ \

t^y . 2 : 2,2 - It—-2 - 1

J sin*e

w

2 2

2(o cos 9 + a., sin 9)co v o 1 2^2

°l

+a

2

From (23), (24), and (25), we have

iB

2

(e

2

B

2 - m

2

o)2

) iB

2

(u>2 - co2

) O O 1 o

v 1

2 , 2

a

l

+ °2

2 3 n m m.oo e e l

nmw'

e e

(32)
°1 iB

2

o 2 , °1 +

2

°2 n m.oo e i

°2 B

2 . °1 +

2

a

2 n e

e

(33)

(34)

Eliminating the conductivities, (31) becomes

1M

e2

B

2

(l + cos

2

6) -

1- sin 9

m m.u) e l

Wi 2n2 2^ 4e B cos 6

o2 2

m co e \ m m.oo /

sine

e l

00 iB (oo. - oo J cos 6 iB sm 6

o l , o nmtir

e e n m.oo e l

co (35)

Divide numerator and denominator by i/i, introduce the plasma frequency,

2 n e 2 e *

oo = , and the Alfven velocity, V. =

p em '

B

A

, and use c = to

Van m.

e l

V/ue

obtain

2^ 1 + cos e

k

2

22 . 2„

co c sin 6

P A

/, 2 .2L" / 2 2\2 , „ /4oo cos e , /., , co c \ .4,. \ 2

+ 1+ 2? SUie

1 w

i V Va/

2

oo 2vf H

2 2 2

\ 2^ co c sin 6 h° w

p

v

A J

(36)

CO c, Though p p

« 1 in the problem under consideration, these terms are

oo V7

P A

retained to allow propagation across the field to be considered. If

. IT 6 is not near -x> these terms may safely be neglected. This is a general dispersion relation 'which will treat hydro-magnetic wave propagation in those regions of the magnetosphere with heights between 10,000

and 50,000 km, for frequencies around 1 c/s. The approximations used

will be justified in the next section. The approximation oo « co. was
not made, as this does not hold above 20,000 km.

To obtain the classical Alfven modes, we now make the approximation a) « co.. Then (36) becomes, provided is not near -z —x— Alfven or slow mode

v

2

, - 2- 2 2 4- 2

COS e k _ 1 + cos - sin 6

2 " ' rf< - 2- "

(37)

A I —-x modified Alfven or fast mode

For low frequency waves the Alfven mode is guided along the field lines,

while the modified Alfven mode has a isotropic phase velocity. As the

approximation co « co. is not a particularly good one, it will be necessary, in general, to use the more complete dispersion relation (36).

For longitudinal propagation, 0=0, (36) yields for the phase

velocity

i=f Si) 2 = ih^ (33)

where the upper sign corresponds to the Alfven mode and the lower to

the modified Alfven mode. It is evident that the Alfven mode will cease

to propagate when co > co. . The group velocity will be given by

v [l - ^V/2

, Al + co. j v =^ = _A iZ wV /i-£L\ (39)

g dk _co_ ~

va( + co.

' 1

+ 2co. V l

Both the phase and group velocities of the Alfven mode decrease with

increasing frequency, while the modified wave does the opposite.

For transverse propagation, set = -r in (36), and obtain
COk

XCOC .,,. ' Alfven

(OP

(40)

V. modified Alfven

A

An interesting result is that the Alfven mode degenerates into an evanescent plasma oscillation for this direction of propagation. The

resonance angle for which this mode ceases to propagate is given by

B)

coV

;an

2e

r =-f-§(:i -—I (a)

CO c .

No resonance exists at this angle or any other angle for the modified

Alfven mode, since the numerator in (36) goes to zero at this angle

when the minus sign is chosen. By applying Lt Hospital t s rule to (36),

k

2

it can be seen that lim —^ remains finite and non-zero for the modie-»e co fied Alfven mode.

The polarization of these waves when 6=0 can be obtained from

the y-component of (28)

2

ico/ua

2

E + (k - icojuo )E =0 (42)

Substituting for the conductivities from (24) and (25), we have for

the electric polarization in the x-y plane

iE eB (e

2

B

2 - m2

co2

)B k

2

co. (to2 - to2

)vfk

2

x o o i o _ _x

v i A /. ?

n E~m.co~" 23 "co" 3 K)

y i n m.eco co.co e i i

To determine the polarization for longitudinal propagation, substitute

(38) into (43) and obtain

+ 1 (44)

iEX

CO.1

CO

2 2

to. - CO CO. X 1

CO. + CO X -

E

y

co(co. - co) to to

10
Again the upper sign refers to the Alfven mode, and we can see that

it is left-hand circularly polarized. The modified Alfven wave is

a right-hand circular wave.

In general these waves will be elliptically polarized. This can

i(E cos e - E sin e) X Z be seen by calculating - , which will give the elec7

polarization in the plane normal to the propagation direction. From

the z-component of (28) obtain

2 2 2 - k E sin G cos 9 + (k sin 6 - ico/uo )E =

o z (45)

Combining this with (42), (23), (24), and (25) we have

i(Ev cos 6 - E sin 6) x z _ E

CO. 00 COS 6

1 p

3 2fk

2sin2

6

(46)

CO

CO c + -

CO 2 2>CO c /

2 2 2,2. Introducing the general dispersion relation (36) and using to c « co V?",

p A

wherever it can be made without losing the ability to treat transverse

propagation, the right side of (46) becomes

co. co VTcos

i p A

(, Cco \ . 2^ + | 4co cos 6 , f-, , co c V 9 TT" ( W

2 2\2

L h ^\ sin46

^p A>

3

2'

CO c

. 2. + 4co c os , /, . a) c \ sin© -A 2— + 1 + -TT ^

w

i \ Va/

. 4 sin 6

2V

2

co2

2„ , _a_d. sin e +

2 2

CO c 1 -

; lcos 6|

CO.'

(47)

IT If one stays away from = —, this rather complicated expression reduces

to

i(E cos 6 - E sin 0) x z E

CO

2co cos (

I . 2 +

sin -i

2 2

4co cos . .4^ ii ,

,

.

- x + sin | (48)

CO. )

11
Both (47) and (48) reduce to (44) when 0=0. When to « to., (48) shows

that, as long as 6 is not near 0=0, the Alfven mode is linearly polarized in the z-x plane, while the modified Alfven or fast mode is

linearly polarized in the y-direction.

For transverse propagation it can be seen from (47) that both modes

are linearly polarized in the y-direction, though it should be remembered that the Alfven mode does not propagate transversely.

Since one often works with magnetic records it would be desirable

to express the polarization in terms of the components of the magnetic

perturbation vector instead or the electric vector. The transformation

-* -> -+ can easily be made by means of k X E = cob, from which it follows that

tob E cos © - E sin © = -r^

x z k

and

to(b cos © - b sin ©)

T-, v z E = - * ; 7 k

Thus, the electric vector in the z-x plane is associated with the magnetic y-component, and the electric y-component is associated with the

magnetic z-x component with the vector aimed in the negative x-direction. With these changes all the preceding statements on polarization

apply to the magnetic perturbation vector. Though E has a component

parallel to k, b does not.

3. The Effect of Ion-Electron Collisions

It is possible that under 20,000 km the ion-electron collision

frequency, v., may approach the wave frequency, to. Thus, it is desirable to develop a technique for handling ion-electron collisions, when

12
their frequency is comparable to the "wave frequency, but still considm v. e le ering& co » and v. « to • m. le e i

The effects of collisions will appear as momentum exchange terms

in the equations of motion, (7) and (8). If we assume that m « m ,

the rate of momentum exchange between electrons and ions is given by

m v. (v -v.) and the equations of motion become

- iam v = - e(E + v X B ) - m i>. (v -v.) (49)

e e

v e o e ie e i' v '

- iam.v. = e(E + v. X B ) + m v. (v -v.) (50) l i

v l o

/ e ie v e i' v '

or approximately,

m (v. - ico)v = - e(E + v X B ) + m v. v. (51)

e

v ie ev e o' e ie l

v '

- im.ajv, = e(E + v. X B ) + m v. v (52) l i

v lo'eiee t v/

The z-components of the above equations are m (v. - io))v -icv. = - eE (53)

e

v ie ez e ie iz z

v

- m v. v - im.cjv. = eE (54)

e ie ez l iz z

v '

Solving for the v *s,

ieE

V

ez = " m (0,+ L. J (55)

e

v ie

ieE

V

iz =m.(o>+ L. ) (56)

i ie

Then

in e e /, . \ in e E

J = n e(v. - v ) =

e

z e

v iz ez 0) +

il„ a -. \ in e E

z /l +

1 L e z /r„\

ii>. Im. m / m (u) + ii>, 7 ie\ i e' e

v ie'

13
Thus,

in e

e (58)

o m (to 4- lv. ) e

v le

Similarly, the x- and y-components of (51) and (52) can be solved

for v , v » v. , and v. . From these J and J . and hence a., and

ex' ey* ix* iy x y' 1

a», can be found. However, if the above approximations are used, it

will be found that o, and o

? are not changed from (24) and (25). Hence,

the main effect of introducing ion-electron collisions is to replace

m (co + lv. )

, ej iei

m by . e

J

co The basic dispersion relation (36) now becomes

1 + cos e -

2 2 I to(to v + lv. )c sin 6,^,2 2_ ie_ +\]4to cos 6 , 2172 "1 2

to V7 co. P A i

1 +

co(co v + ip. )c ie'

2-,2

CO2-L

siiTQ

co 2V

2

A

\ co.

2 2-i

2\ _ co(co + ii>. )c sin 6

2^ _J le' cos e -

to V7 Va (59)

The actual effect of ion-electron collisions on the dispersion

IT relation is very small. Provided 8 is not near p, we can use the ap2

proximation

co(co + lv. )c'

! le

COP A

2 2

CO c « 1, and obtain the same dispersion relation as approximating

2

-= « 1 in (36) would give. In the case of

co V7

P A

transverse propagation the modified Alfven wave is uneffected, but the

Alfven wave is given by

,2 ico / lv. \ co p

co2 CW \

2t0 / 2cco

ICO 2 ceo

to \ ~ / 2cco

The propagation constant now has a real part and propagation will be

14
is possible where becomes large enough, though attenuation will be

significant. Thus, the only place where ion-electron collisions play

a significant role is in the transverse propagation of the Alfven mode.

The dispersion relation (59) bears some resemblance to the Appleton-Hartree formula for radio frequencies, especially as to the role

played by collisions. However, the approximations used in deriving

the respective equations are different, as the Appleton-Hartree relation is for much higher frequencies than (59)«

4. Justification of Assumptions

It will now be worthwhile to tabulate the various plasma parameters and justify the assumptions used in the preceding sections. Four

models of the magnetosphere will be considered, corresponding to day

and night, and maximum and minimum of the sunspot. cycle. The basic

data has been taken from Prince and Bostick [1964].

In order to derive a dispersion relation for the Alfven modes,

-> -* ->-*-» -» jJtz • r it was necessary to assume that b, E, J, and v vary spatially as e ,

which implies that the plasma is uniform. A reasonable criterion for

uniformity is that |k| must not change by more than ten per cent within

one wavelength

ig| ds X |ds|-

where V is the phase velocity, s, the path length along the direction

of propagation, and f , the wave frequency.

The first case to be considered is that of wave propagating radially in the equatorial plane at the Alfven velocity. Here the

15
requirement is stated simply as

-af |<o.if

where V is the equatorial Alfven velocity and r the geocentric height,

Figure 1 shows the magnetosphere divided into uniform and nonuniform

regions. As the frequency becomes higher more of the magnetosphere

can be considered uniform. For frequencies greater than 45 c/s, all

of the magnetosphere above 500 km is uniform. If the wave period is

greater than 100 sec, none of the magnetosphere can be considered uniform. Off the equatorial plane the Alfven velocity will be increased

by the factor in. + 3 sin A., where A. is the geomagnetic latitude, and

the severity of the restriction will be increased. The restriction

will be relaxed if the propagation direction is different from that

of the gradient of the Alfven velocity.

A more realistic type of propagation is that of an Alfven wave

traveling along a field line. If the field is assumed to be a dipole

field, the equation of a field line is

r = r cos A. e

Then, if s is the arc length along the field line — =t/l + r

2 — 2 = — + 3 sin2^ dr " y dr 2 sin A. A

and

dV

A dr d\ dr /— .2, dV

Ae , 3VAe

sln ^

cos *

dX -3— = -r- 3— = -^Vl + 3 sin X— 1 — ds ds dr ds dr / ~— dr

A + 3 sin A.

16
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a

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fHo

COao

•H

hO

0)

fn

O

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I PiO

Pi t3

o

O

+3

Pi •H

n3

a> <d•H

i> •H

nci

(U

fH

(D

fit ft

coO

-P

CDPi

CD EH

pa&-p

co •H

-P

o

<+H •H

Pi

Pio

<+H

Pio

•H

fH

CD P•H

fHO

CU EH

H

ft Ha3 •H

fHo8

&

cu

cu P

Pi H

•H

T3

a3

fHbO

Pi •Hs

M

n3ftO

fHft

.ap

bD

Pi

cu Hcu

is

Pi p

cua

fH

cuft oH

P

co

co

cu H

>n>

cu

,3oH

a3

to

CO lO rO OJ

(UJ>1) 3DVddnS 3A09V 1H9I3H

H

17
dV. dV 3V sin X cos

2

\

-r^ = 2 sin A. --— — = < 0.1 f

ds dr /., . . 2. > r(l + 3 sin \)

The boundary of this restriction is shown on a polar plot (fig. 2) for

several frequencies. Near the equator the boundary is closer to the

earth, since the field lines there are almost perpendicular to the velocity gradient. But toward the poles the field lines are close to

being parallel to the velocity gradient and the gradient itself is greater,

which pushes the boundary away from the earth. The plot is to be interpreted as representing the inner boundary of the uniform plasma for

a wave with the specified frequency. The values used in figures 1 and

2 are given in table 1. They are an average of day, night, sunspot

maximum, and minimum conditions, and are merely typical values. Though

the AlfvSn velocity at a given point may vary by a factor of ten, the

change in the plots will be relatively small. At any rate, it is clear

that the assumption of a uniform plasma for waves with frequencies aground 1 c/s is not really valid over much of the magnetosphere. It

is necessary to make this assumption in order to derive a dispersion

relation and the polarization properties for Alfven waves, but it should

not be surprising that these waves show observed properties which cannot

be explained by, or are inconsistent with, the simple theory derived in

the foregoing sections. In particular, reflection and refraction of

energy would be expected if the plasma parameters have large gradients.

The assumption of a collisionless plasma depends mostly upon the

ion-electron and electron-electron collision frequencies, which in turn

depend on the electron temperature, a quantity which is largely spec18
50,000

e

LUO

<

Ll. (T

CO

LU

>

o

CD

<

X

UJ

40,000-

30,000-

20,000

LU

_lO

Q.

„ EARTH, EQUATOR —• 1 6

GEOCENTRIC HEIGHT (EARTH RADII)

Figure 2 - Uniformity conditions for a hydromagnetic wave propagating along a field line. The plasma beyond the curve for a given frequency may

be considered uniform for that frequency. A dipole field is assumed.

19
TABLE 1

Typical Values for the Gradient of the Equatorial Alfven Velocity

Height

(km)

Geocentric

height

(km)

VAAe

(km/sec)

Ae

dr

(sec )

r

(see"" ) 1000 7370 2020 + 3.80 0.274

1500 7870 3570 + 2.00 0.454

2000 8370 4020 + 0.437 0.480

3000 9370 4200 - 0.156 0.448

4000 10,370 3720 - 0.469 0.359

5000 11,370 3270 - 0.418 0.288

7000 13,370 2490 - 0.330 0.186

10,000 16,370 1620 - 0.178 0.0990

15,000 21,370 1060 - 0.0822 0.0496

20,000 26,370 799 - 0.0357 0.0303

30,000 36,370 526 - 0.0217 0.0145

40,000 46,370 366 - 0.0127 0.00789

20
ulative. Nicolet [IS63] gives the ion-electron collision frequency

as (r\n ,-6

x 10

59.1+4.-18 ' *'* ** tr?z- e

when

p x m » 1. The great problem here is deciding what T should be.

ene If we assume that it is the same as the ion temperature, and that this

is 1500 K at sunspot maximum and 1000 at minimum, we obtain the results displayed in table 2. But there are reasons to believe that the

actual electron temperatures are much higher than what are assumed here.

Chapman [I960] believes that the solar corona plays an important

role in determining the temperature of the upper atmosphere. Best estimates of the coronal temperature at the distance of the earth from

the sun are between 50,000 and 100,000 K. With conduction of heat

downward, a smooth transition from coronal to ionospheric temperatures

would be expected. It is not unreasonable to expect temperatures in

excess of 10,000 K above 10,000 km. This would reduce the values in

table 2 by a factor of at least 20 and certainly make the collision

frequency negligible above 4000 km.

According to Liemohn and Scarf [1962] evidence from cutoff frequencies observed in whistler data indicate temperatures on the order of

100,000 K in the outer magnetosphere. Their calculations are based

on the assumption that the cutoff is controlled by thermal broadening

of the cyclotron resonance.

ocarf [1962] also used the cutoffs observed in micropulsation data

21
TABLE 2

Ion-electron Collision Frequencies Based on the Low Temperature Model

Height

(km)

Geocentric

height

(earth radii)

Day Max

(sec-

)

Day Min

(sec )

Night

Max

(sec""

1

)

Night

Min

(see" )

1000 1.16 75.2 27.6 20.6 6.96

1500 1.24 22.0 11.4 8.96 3.84

2000 1.31 12.4 7.47 6.19 2.93

3000 1.47 7.27 4.38 4.15 2.10

4000 1.63 5.02 3.07 3.22 1.63

5000 1.78 3.86 2.35 2.60 1.32

7000 2.10 2o50 1.52 1.77 0.932

10,000 2.57 1.44 0.876 1.02 0.621

15,000 3.36 0.683 0.413 0.483 0.291

20,000 4.15 0.353 0.215 0.249 0.153

30,000 5.71 0.125 0.0759 0.0884 0.0536

40,000 7.28 0.0601 0.0365 0.0427 0.0264

50,000 8.85

J 0.0349

| 0.0212 ; 0.0246 0.0150

22
to support the existence of high temperatures. He argues that the thermal

motion of the ions will have a Larmor radius associated with it, which

will determine the smallest wavelength which can propagate. Since few

micropulsations are observed with frequencies higher than 3 c/s, temperatures of 10,000 to 100,000 K would be expected.

These conclusions support Chapman 1 s hypothesis that heat is being

conducted downward from an extended solar corona to the ionosphere.

More commonly, the upper magnetosphere or exosphere is assumed to be

isothermal. Above 500 km disassociation and recombination is assumed

to be negligible and diffusion is the most important process at work.

Evidence from various sources indicate the temperature at the 500 km

level to lie between 1000° and 2000° K [U. S. Standard Atmosphere, 1962;

Prince and Bostick, 19643* Hence, under the foregoing assumptions these

temperatures would be expected to pervade the exosphere. The crux of

the argument lies in deciding the importance of conduction from a hot

solar corona.

Even if the ion-electron collision frequency approaches the wave

frequency, the results of the last section indicate that the theory

is not greatly affected by its inclusion.

Some comment should be made about electron-electron collisions,

or in other words, the electron gas pressure. In the first chapter

of his book, Stix [19623 discusses briefly some of the effects. To

use one his results, pressure terms will not be important if

ne

2

V? -2-^«i

xT a) e p

23
where k is Boltzmann's constant. This condition is easily met.

Table 3 tabulates the various parameters appearing in equations

(36) and (46) for the equatorial plane. B , go., go , t/go go., and V should

O _L G " J- }\. be increased by the factor vLl + 3 sin X for locations with latitude A.. 8.1 x 10

B was calculated for a dipole field, B = — «

, the ion gyrofrer

eB

o 7

B

o quency from go. = —rr = 9*66 x 10 77-, where M is the mean molecular

weight and m-, is the mass of a unit molecular weight. Above 2000 km

M is always 1.

If go is on the order of a few rad/sec, it is clear that go « go * e and go « "/go co. is always satisfied. The latter inequality also guar2 2 vVV~gj

, . , , GO c .. .. . A p antees that « 1, since go go. = —

1-. 2T72

' e 1 2

go V7 c P^ A

m v. e ie In section 3 we used the approximations « go, and v. « go .

1

m. * ie e 1

The largest value of v. appearing in table 2 is 75*2, which means that

ie

e 1

m v

< 0.04 rad/sec for heights above 1000 km, which is much smaller

m.1

than the wave frequencies of interest. Comparison of tables 2 and 3

supports the second inequality.

6KT

To justify p -1/0 >> 1 in "the derivation of the ion-electron col- _e

; n"^

1 o lision frequency, note that the smallest possible value of T is 1000

10 -3

K, and the largest value of n above 1000 km is 8.2 x 10 m . Thus

e*T 3 e »10 and the condition is met.

e n

e

2k
nO nO •o nO NO nO nO NO in in in in

m

C rH o UA o NO ON -4 rH -4"

O nO to to

•H ir\

m cv to nO -4 CV o to o to ON -4" -p g • • • • • • • • • • • • •

•H

cn CV cv rHH rH

HH nO -4" cvH

H

nO "no"" nO NO NO NO nO nO nO WA IT\ ir\

m

r—s S S 3 m CV

O nO cn CV in On ir\ >n

H to

erj u

en -4 NO rH to en

J

rH -4- ON -4m

g • • • • • • • • • • • • • CO ir\ -4 en en cv cv rH H to -4" en cv

vO sO nO NO NO nO nO NO in

m in in >n

C UA cn -4 nO o O o o to -4 cv

m to

a, •H

H

m nO ,^_ cn o NO cv

H to -4" en £N- 3 a > -4 cn cv cv cv H rH to liA en cv rH O- £> nO NO NO NO NO NO nO nO »n in u> X nO CV -4 en to ;— cv On cv H cv c>- to

03 to UA en to o -4 to o -4

O o OO

g • • • • • • • • • • • • • cv H o -4 cn cn cv cv rH H in -4 cn

vO

O nO NO NO NO NO NO NO nO in in UA c NO to -4 o- to o o <f -4 O NO cv -4 P

bO •H

•Hg

vO to in UA ON -4 rH cv -4 rH cv

O £n- -4 C-- O nO 1^1 in -4 cv rH H £s_ in cn

U"\ nO nO nO NO nO nO NO W\ in in

m in

CO S a

-4" nO o nO rH "•

to ON -4" to c- cv NO

u MD

H cv en to en cr- cv in cn rH o rH

V •—S.

g -p

0) "1

c*^ rH cv en cv cvH rH t» NO -4 cv cv ! NO nO NO nO nO NO nO NO nO ir\ in UA in U d to O NO £•«. in nO to r>- -4 -4 to in in

CtJ < •H "V

m NO en to en ir\ to CV CV o cv

H CU > g • • • • • • • • • • • • • o

CM -4 -4- -4 cn cn CV rH

H ON NO -4" cn

•H UA ir\ NO NO NO nO NO nO l^ UA in LT\ in

Sh 9 £

CV o cv -4 -4 ON to in -4"

H in r-i

CD en M3 in cv ON -4 o H en in -4- tc cn x; g a cn r- rH cv CV H H H ^» >n en cv H 8 § • m +3 •—N

-4 -j- <r -4 -4 -4 -4 en cn en cv cv CV <J Q> 1- c O UA o NO NO NO ON ON H cv NO in •

6-1 B. •Hg

-4-• to•fN_• o• • CV• en•

IP*• -4• to• • en* g U m nO UA -4 CV cv rH c- en rH NO en rH S

•H -4 -4- -4 -4 -4 ~4 -4 en en cn cv cv CV

g 3 W

O

O o NO NO NO ON ON

H cv nO

m t>- o a; ri H H -4 o o cv cn IP\ -4" to On en to

CO .3

g • • • • • • • • • • • • • > CV CV en -4 cv cv H ^_ en H NO en H

<D XT

nO nO -o NO NO UP* in in in -4 -4 -4 en

ua CV en cn £^- cv cv -4 NO -4 O- en Is- in 3 aJ ITS o -4 £> cv NO o CV -4 CN- ON -4 ON" 1 Us""' CA cv CV H rH ON in cn H 2s" CV H Cn- rH

^-v o o ON in en -4 to

M * C to o en H to to in to • • • to cn cv

•H 7? no en in

O cv cv £> ON CV NO • • o g to «H rH o nO in en rH J

-4H o- -4 •

a)U

(V

n—' On in en -4- tO (0 a

-4 o UA rH to to

m to • • • to en

C

•H cv ua C- tf\

O cv cv >*«_ ON cv NO • • CfJ 3 g I-i H -4 o NO in en rH CN- -4- H f^ ? 8 ^

in o < m UAi in NO nO1 NO nO nO r- c>- J

to CO 1 i cv 1sO ito -4" 1 cn i nO 1 -4- 1 1 O 1 ON 1 cv 1 cv en o PQ g o •nO en to cv -4 en to cv -4 NO H UA • • • • • • • • • • • • • • cv s-x cv H H o £>- UA en H to -4" H to -4

-P o O

O o o O o O o O

O o O -P -S.O o O O o o O o O o o o o bo a 8 £ H 5 Q m O o o o o o Q o o o H rH CV cn -4 wn £•»_ »> •V «\ •\ •> •\ u ^ o in o o o o -c H H cv en ~4 in

25
The assumption, —-—p— « 1, allowed electronic pressure terms

T <d e p

to be neglected. Using the largest possible Alfven velocity and the

V2v

A -8

smallest temperature, we find that p— < 4 x 10 • T co e p

At the beginning of the derivation, the displacement current was

v2

A

neglected, which is equivalent to stating that —x « 1. To see this,

c

note that we want —— « 1. From (26), |kE| = |oojuj|. Thus we need

V

2

2 V7

~~2~ « 1 or -^ « 1

k c

5. Propagation in a Multicomponent Plasma

So far the treatment has only considered a plasma consisting of

one kind of ion. When several species are present, it is valid to conEn.m.

sider all the ions as having an average ionic mass, m. = — , provided

i

the approximation to « co. can be made for each of the ion species.

Fortunately for geomagnetic micropulsation studies, ions heavier than

hydrogen exist only below 3000 km and in this region all the ion gyrofrequencies stay above 100 rad/sec, well above the normal range of micropulsations . Though micropulsation theory is not concerned with multicomponent propagation and the subject has been treated in depth by

Smith and Brice [1964], some of the simpler aspects of the theory are

included here because of the ease with which the foregoing treatment

can be extended to include multicomponent effects.

26
Consider the plasma to consist of electrons and j kinds of singly

charged positive ions. Negative ions and multiply charged ions are

rare in the ionosphere and can be neglected. The same dispersion relation (31) "will still be valid, but it will be necessary to calculate

new values for the conductivities. With the realization that equation

(8) now represents j equations, one for each of the ion species, the

same equations of motion can be used. The definition of current density must now be written as J = e(En.v. -nv ) and the neutralitj?" condition becomes n = En. . Throughout this section all summations will

be over i and will run from 1 to j, The approximations to « to and

to. « to will still be used.

1 e Following a procedure similar to that used in section 2, we obtain

2

. 2 / n. n \ in e a = i£-fc>i + -£ ) « -2- (60)

o to \f-Jm. m / m co

v Vie/ e

2 / n.m. n m \ „ n. °1 =

"

ie W(Z 22

1 1

2 2

+

2 2

& 6

2 2j~ " ie aE ; 2 2,

(6l) \eB-tom. e B - 10 m / m. (to. - toc )) o e 2. 1

and

°2 " AlS 2/ 2 2 " v"

6

2 2Vfe-rH < 62 > \ eB -torn. eB -com/ o to. -to

o 1 o e 1

Making use of the identity

(n.to. v<i2 , n.to v-i n. n.

to. - to / \ to. - to / 1 i 1 we obtain

n 2 2 n. n.

<& . /c e to v^ 1 v^> 1 / / . \ ai

+ °

2 = " ^^r^r^rr^

B 1 1

W o

27
If we substitute for the conductivities in (31) and divide the numerator and denominator by

lco B

„ n.

2^--% i_

oZrf / 2

m. (co. - co )

n. . n.l CO)/ / A—

+ CO 1 X

we obtain a new dispersion relation, equation (65) on next page.

By introducing a composite ion gyrofrequency defined as

n. . n.co. OtLu i 2 2 N / j 2

eB^

m. (oo. - co ) ^—^oo. - co n = hr1 h (66)

E_\ y^ n

i

2 2 S-J 2 2

CO. - CO *^^C0. - CO 1 1

and a new Alfven velocity defined as

n. B Y* -

oC-u Lu ,t 2 2 N m. (co. - co ) VT = ±-_i (67) A ,^_, n. n.

v

^-^CO, — CO^-^LO, + CO I

the dispersion relation (65) can be put into the same form as (36)

t _i 2^ co c sin 6 . |4co cos 6 , /-, , (o c \ .4^ 1 + COS 6 - 7T-= h -4/- = h H—;=—7T 1 sin

2 coV V,

2 2 / 2 2 v tz

JS- = E-A ! : E-£ (68) 2

2V

2

(-S

2V 2 2 . 2j 2_ co c sui

cos G -

oj"V?

P A

With the above substitutions and some algebraic manipulation the polarization equation

2 i(E cos 6 - E sin Q) ±o cos e(k - ico/xo — ) ~ E—" =

° 2 2

" < 69 > y °2^ s^n e ~ lt0 'u:r )

28
^1 3

29
can be put in the same form as (46)

i(E cos 9 - E sin G) p X Z ' CO J

^

3 2/ k

2

sin

2

w C

6 J^

I 2

+ 22

\ (0 0) c

1-11

- ,

(70)

)

Thus, all the dispersion and polarization properties for propagation

in a multicomponent plasma can be obtained from the equivalent equations

for a single component plasma simply by making the substitutions (66)

and (67) for to. and V. respectively. However, formulas which involve

taking a derivative with respect to frequency, such as the formula for

group velocity, will not carry over, as V. and 12 now are functions of

co. It can be seen that for j = 1, (66) and (67) reduce to co. and the

classical form of V.

.

A

For low frequencies, co « to.

,

2 2

B B

V

2 = a - o

and

A En.m. „

1 1 ^Nm.

eB En.m. eB m. Q _ o 1 1 _ 01

En.m? 2 11 m.1

where N = En.. Since at low frequencies, co «Q, it is sufficient to

account for multiple ion species by simply using the average ionic mass

in the single ion equations. As co approaches any of the ion gyrofrequencies, 12 becomes equal to that gyrofrequency.

Since Q may be negative the upper sign does not necessarily go

with the Alfven (left-hand) mode as before. This point may be clari30
fied by examining the polarization at ft = co, a point known as the crossover frequency. When (48) is translated into multicomponent form, it

becomes

i(Ev cos 6 - E sin e) n x z » E 2a) cos 7

L±n2e t ^E|sfe ~

3

. n

4

e j (?1)

which is indeterminate as both Q -* oo and 6 -» 0. (Letting co -* would

produce the same problem, except that waves of zero frequency are uninteresting.) What is happening is that the angle at which polarization changes from circular to linear goes to zero as ^

-* 0. If sin 6

2 2

« , the polarization is essentially circular and the propagasr '

~

tion is considered quasi-longitudinal. But if the reverse approximation holds, the polarization is linear and propagation is quasi-transverse [Booker and Dyce, 1965] • Paying close attention to signs, the

polarization for 6=0 becomes

7 ' '

Thus when ft > 0, the upper sign will give left-hand polarization, but

when 9. < 0, the lower sign must be taken to obtain the same mode. A

consequence of the sign interchange as co goes through the crossover

frequency is that the two modes will interchange their wave-normal surfaces. The faster mode is always the isotropic mode. If, because of

changing plasma parameters, a wave propagates through a crossover point,

it will interchange modes rather than exchange wave-normal surfaces.

This point will be made clearer in the ensuing discussion.

31
For longitudinal propagation, 6=0, the phase velocity is given

by

V

p = £

= V V^1 -

B

° (72)

to.i

From this point on the upper sign will refer to the Alfven (left-hand)

mode, regardless of the sign of Q.

V^ n

i

Resonances will occur wherever > = °o Since to > in this

Z--/^ ^ to

treatment, only the Alfven mode produces resonances, and these will

occur at u = to. . (The other mode has a resonance at to = to , but this

i

v e*

result cannot be obtained from (68) as the approximation, to « to , was

n.i

e' used in the derivation. ) A cutoff will occur where / = 0, *-* i +

and again only the Alfven mode will have cutoffs in the frequency range

V^ n

i of interest. Since > — = is an equation of order j - 1, there

will be j - 1 cutoffs. By noting that V is real as to approaches to.

from below, and imaginary when to is just above to. , it is clear that

a cutoff must occur between each pair of ion gyrofrequencies. At the

crossover point, determined by Q = oo, both modes will have the same

phase velocity when propagating longitudinally. This will happen wherE

n

i

~2 2 = ®> and one such point will occur above each cutoff,

to. - to l

Figure 3 shows these critical frequencies for a plasma with three

types of ions. Propagation can occur wherever the phase velocity is

32
\1

U)

(A)

1 Figure 3 - Phase velocity for longitudinal propagation in a plasma with

three ion components. (co„ = 2u), , co_ = 3^-. , n

? = n,, n~ = O.ln,,

VV = 2—.)

A1 ; ' ; v/un^m..

33
real, which for the Alfven mode is for frequencies up to the first gyrofrequency, then from the first cutoff to the second gyrofrequency,

etc., until the last gyrofrequency is reached. The effect of letting

the number density for one species go to zero is to move all the cutoffs toward that ionT s gyrofrequency, which narrows the passband controlled by that ion.

Another significant set of frequencies is generated by the condiEn.o).

—

p = 0. At these frequencies the Pedersen

co. - u> 1

conductivity is zero and no current will flow in the direction of the

transverse electric field. There will be j - 1 of these zero current

points and one will occur below each of the cutoffs. Here only the

modified Alfven wave will propagate and it undergoes a reshaping transition at this point, the wave-normal surface changing from a sheroid

to a dumbbell lemniscoid.

Table 4 summarizes the types of significant frequencies arranged

in order of increasing frequency. The last column indicates the type

of transition in the notation used by Stix [1962]. The entire cycle

will be repeated j - 1 times. The situation can perhaps be best displayed in a diagram which shows the effect of the various transitions

on the wave-normal surfaces (fig. 4). At low frequencies the surfaces

are as shown in the upper right corner, with the Alfven wave field guided.

As the frequency increases, we proceed around the diagram in a clockwise direction. At the resonance, L = °°, the Alfven mode disappears.

At the zero current point, S = 0, the modified Alfven wave is transformed

34
TABLE 4

The Significant Frequencies for Propagation in a Multicomponent Plasma

Determining

Condition

Type of

Transition

resonance

zero current

cutoff

crossover

CO = tO.

*—* n.to.

fc «to. - to 1

/ J lti- - to

/ *to. to. - to

L = oo

S =

L =

D =

35
8oii

CO

(T

Vu c

01 >;,

(1) rH O CO3O

•HP

C!Oo oc

a>3. °

(0 £ 0) o o

-Pu

co

mapc©coIo

•HPrHI

a)uo

«h

to<DO

05

CO o « P

a) *-< O H

ft) -H 3 cm

CTV oHP

<D&

n3•Co

•H

(0

to

0)u

a no 3 Q P

a5!

CO

to

f-. av

co !ooO

•H

Cm

36
into a field guided wave. The Alfven mode reappears at the cutoff,

L = 0, as an isotropic wave. Finally, at the crossover or D = transition, the surfaces become tangent and they interchange their polarizations. After completing the circuit j - 1 times, the final resonance

is passed and only the modified Alfven mode is present, as shown in

the lower right corner.

Table 5 specifies a six component (H , He , N , , NO , and Op)

model ionosphere. The number densities are averages obtained from the

papers cited at the bottom of the table. The resonances, cutoffs, and

crossovers for the Alfven mode were calculated as a function of height

and are displayed in figure 5« The shaded areas delineate the regions

in which the Alfven mode can propagate. The resonances, located along

the right edge of the passbands, depend only on the field strength and

the ionic mass. The cutoffs, in contrast, depend greatly on the relative number densities of the ions. For example, it is seen that N ,

which is relatively scarce, controls a very narrow passband. The crossover frequencies, which are shown by a dashed line, are located in the

shaded regions, but close to the left edge or the cutoff frequencies.

The existence of a crossover point at any height acts as a block

to the transmission of a modified Alfven wave through that height.

Such a wave propagating downward through a crossover point will be transformed to an Alfven wave and then reflected at the next lower cutoff.

Similarly, a modified Alfven wave attempting to propagate upward through

a crossover point will be stopped by a resonance. Thus, a major effect

produced by a multicomponent plasma is that a modified Alfven wave with

37
TABLE 5

A Model Multicomponent Ionosphere

-

height

(km)

H

+

n

l

He

+

n

2

N

+

n

3

+

n

4

N0

+

n

5 n

6 (nf3

)

150 2.4 x 1010 2.2 x 10

11

1.5 x 10

11

200 3.0 x lo

11 2.0 1.2

300 6.0 x 10

10 12

1.1 x 10

400 1.8 x 10

8

1.2 x 109

5.5 7.0 x 10

11

500 8.6 2.9 4.6 4.4

600 1.2 x 109 3.6 3.5 2.7

700 1.6 3.9 1.8 1.6

800 1.8 4.0 3.2 x 109 1.0

900 2.1 4.1 7.8 x 10

8

6.2 x 10

10

1000 2.4 4.1 2.7 3.8

1100 2.6 4.0 1.2 2.4

1200 2.7 3.9 5.5 x 107 1.4

1300 2.8 3.7 2.5 8.8 x 109

1400 2.8 3.4 1.4 5.4

1500 2.7 3.0 3.4

1600 2.6 2.7 2.1

1700 2.5 2.2 1.3

1800 2.3 1.7 8.0 x 10

8

1900 2.1 1.0 4.9

2000 1.8 5.0 x 10

8

3.0

The following references were used in the above compilation: [Johnson,

I960; Hanson, 1965; Nicolet, 1962; Taylor et al., 1963; Nawrocki and

Papa, 1961; Taylor and Brinton, 1961]. Exponents are assumed to carry

down a column until the next is reached.

38
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39
a frequency in the vicinity of the ion gyrofrequencies will be unable

to propagate through the ionosphere.

An Alfven wave, propagating in one of the shaded regions but above

a crossover point, is blocked from traveling upward by the resonance,

but is free to propagate downward. At the crossover point it is converted to a modified Alfven wave and it can then propagate through the

cutoff. Simarly, an Alfven wave generated below a crossover point can

travel upwards, but not downward.

40
PART II

The Properties and Interpretation of Pc 1 Micropulsations

A reasonable hydromagnetic model for the propagation of pc 1 micropulsations is one in which Alfven

(left-hand) waves travel along field lines to reach

the earth's surface at auroral latitudes and then

propagate equatorward as modified Alfven (righthand) waves just above the ionosphere. The isotropic

right-hand wave is constrained to a duct between

400 and 2000 km above the earth by refractive processes. It is shown how many of the observed properties of pc l*s can be related to such a model.

Mechanisms for the generation of pc l*s are also

discussed and it is concluded that particle instabilities are the most likely source.

1. Introduction

One of the more interesting classes of geomagnetic micropulsations

are the pc 1 or pearl pulsations, which have periods ranging from 0.2

to 4 sec. On an amplitude chart of sufficient resolution they appear

as a smooth sinusoidal carrier signal with irregular amplitude modulation (fig. 6). The envelope has a tendency to repeat its general shape

over an interval of several minutes. Often the period remains constant,

ignoring minor fluctuations, for the entire event, which can last several hours. When displayed on a frequency-time plot, sucn as a Sonagram, pc l T s reveal a regular fine-structure consisting of a series

of tones, usually rising rapidly (fig. ?)•

For the most part the early work in geomagnetic pulsations was

confined to those of longer period. But often there was reference to

faster oscillations, e. g., Van Bemmelen's [1908] "spasms" and Harang's

[1936] "vibrations'1 . Of particular interest here is a paper by Sucks41
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43
dorff [1936], in which pulsations of 2 to 3 sec period and with the

general appearance of a TT shuttle tT are described. "It is seldom that

such an oscillation occurs singly; usually there is a number of oscillations, interrupted by short intervals, for one hour, for several hours,

sometimes even for more than 24 hours continuously, when the curve often

resembles a pearl necklace consisting of oval pearls of different sizes, TT

To the best of the author 1 s knowledge, this is the earliest reference

to pearls by name. It is interesting that Sucksdorff reported pearls

to be an essentially daytime phenomenon with a maximum about 1400 local

time. Much later pearls were considered to be nocturnal [Benioff, 1960J

Tepley and Wentworth, 1962] on the basis of studies made in middle latitudes. Heacock and Hessler [1962] restated that in auroral latitudes

pearls occur primarily during the day. To return to Sucksdorff1 s work,

he concluded that September was the most favorable month, that no 27

day recurrence pattern was evident, and that though the pulsations were

local in detail, general characteristics might be spread over several

hundred kilometers.

Attempts to relate pc lT s to other geophysical phenomena have at

best met with limited success. Troitskaya [1961] indicated a tendency

for them to precede magnetic storms and to coincide with increases in

cosmic ray intensity. Heacock [1963 ] showed several examples of pc l f s

having rather sudden onsets accompanied by increased cosmic noise absorption. However, it should be emphasized that, in general, no correlation between pc l T s and ionospheric data has been shown. Some authors have stated that pc l*s occur during magnetically quiet periods,

44
but other evidence indicates pc 1*3 to be independent of general magnetic activity. Possibly, the relative ease of detecting pc lT s at

quiet times led to the former conclusion. It has been suggested that

pc l T s are less common at sunspot maximum [Troitskaya, 1961]. There

is, however, evidence that the occurrence of pc l T s is enhanced during

the week following a magnetic storm [Wentworth, 1964a].

Along with other types of micropulsations, pc lT s have been plagued

with a multiplicity of nomenclature [Matsushita, 1963]. The term pearls

originated by Sucksdorff [1936] was used by Troitskaya [1961], who also

introduced the abbreviation PP. Benioff [I960] classified micropulsations into four period ranges, with Type A corresponding to pearls.

Later Tepley [1961] used the term hm-emission to emphasize their hydromagnetic character. The International Association of Geomagnetism and

Aeronomy (IAGA) at their Berkeley meeting in 1963 agreed on a system

of classification which divided micropulsations into regular and irregular categories, and subdivided these by period. The classification

is as follows:

0.2 to 5 sec period

5 to 10 sec

10 to 45 sec

45 to 150 sec

150 to 600 sec

1 to 40 sec

40 to 150 sec

Still other terms have been used such as hydromagnetic whistlers [ObaRegular

;

pc 1

pc 2

pc 3

pc 4

pc 5

Irregular

;

pi 1

pi 2

45
yashi, 1965] and CPsp [Yanagihara, 1963 ]• The author prefers pc 1,

since it uniquely specifies the phenomenon and it has been accepted

by the IAGA. However, the term pearl is still frequently used in the

literature and it should be considered interchangeable with pc 1.

2. Properties of Pc lT s

Many of the most significant properties of pc l f s are best displayed

by a frequency-time plot. The distinguishing feature of a pc 1 is a

fine-structure consisting of tones repeating at regular intervals usually on the order of one or two minutes. Most often the tones are rising,

but sometimes they are vertical and occasionally falling tones are observed. Sometimes a fan-shaped structure is observed, i. e., successive tones show a slower rise rate, and this has been considered evidence of dispersion during the propagation of the wave. It should be

noted, however, that the majority of pc l's do not show such a fan-shaped

structure.

The complexity of the amplitude charts is readily explained by

the frequency spectrum. Generally the tones overlap to a considerable

degree. Thus, at any instant the chart is recording the beating effects

of several tones. It is also easy to see how two stations may have

similar frequency structures and yet have dissimilar amplitude records.

When simultaneous Sonagrams for conjugate points were compared,

it was found that between hemispheres the fine-structure was displaced

by half the repetition rate [Tepley, 1964]. This result has since been

verified over a wide range of latitudes [Yanagihara, 1963 J Campbell

46
and Stiltner, 1965] and appears to be a consistent property of pc l*s.

The probable explanation is that pc lf s are produced by waves bouncing

between hemispheres. In contrast simultaneous Sonagrams from two stations in the same hemisphere show no such shift in the fine-structure.

Tepley reports that the equatorial station of Canton Island sometimes

records a doubling of the fine-structure, i. e., a repetition time half

that of stations away from the equator. This could mean that Canton

occasionally sees signals generated in both hemispheres. Careful measurements indicate that the signals do not occur quite simultaneously

in the same hemisphere. Tepley, Landshoff, and Wentworth [1965] found

that waves reach College, Alaska six seconds sooner than Kauai, Hawaii.

5 This would indicate an equatorward propagation velocity of about 8 x 10

m/sec.

As mentioned in the introduction, pc l*s are essentially a nighttime phenomenon at low and middle latitudes, but occur during the day

at high latitudes. Wentworth [1964b] suggests that this may be caused

by differences in ionospheric attenuation. Wentworth argues that though

the maximum production of pc l f s is probably in the afternoon, higher

ion densities at that time greatly attenuate the waves. Thus, most

pc l*s are seen at night, when the ionosphere is more or less transparent.

At auroral latitudes ionospheric attenuation will be much less and the

waves can get through at the time of their maximum production. In addition auroral absorption may provide partial blocking at night.

Attempts to relate observationally pc 1* s and ionospheric activity

at College have shown no correlation between the two phenomena [Dawson,

1965]. There seems to be no significant agreement between the occur47
rence of pc l T s or their amplitude and various ionospheric parameters,

such as f F„, f . , or riometer absorption. Also, some long events

o 2' mm* '

have been followed through sunrise or sunset with no apparent ionospheric

effects [Campbell and Stiltner, 1965]. Pc l*s at conjugate points often

show similarity even though ionospheric conditions at the two stations

may differ drastically. It would be incorrect to state that attenuation effects are never observed, but it appears to be a minor factor

in determining their occurrence.

All this does not necessarily contradict Wentworth, as he predicted

strong attenuation only for the lower latitudes. It would be worthwhile

to investigate the question of pc 1 ionosphere correlation for these

latitudes.

Though it has been suggested that large auroral zone pc l T s with

sudden onsets are accompanied by an increase in riometer absorption

[Heacock, 1963], the author found little correlation between the two

phenomena at College [Dawson, 1965]. Pc l T s occurred during times of

no, moderate, and heavy absorption with the distribution one would expect from chance occurrence. In general no consistent relationship

between pc l»s and particle events has been shown [Campbell and Stiltner,

1965; Tepley, 1965].

In the auroral zone the polarization of the magnetic perturbation

vector has been observed to be confined to a plane perpendicular to

the magnetic field line [Dawson, 1965 ]. When a similar measurement

was attempted at Boulder, Colorado, the result showed a random orientation of the polarization plane [Pope, 1965 J. Apparently the situ48
ation at middle latitudes is more complex than at high latitudes. The

property of polarization perpendicular to the field line would explain

why it is difficult to detect pc l T s with a total field detector, such

as a rubidium or helium magnetometer.

Attempts to measure the sense of polarization have produced confused results. Both left- and right-hand elliptical, as well as linear,

polarizations have been observed. Indeed, the sense will often reverse

itself in the middle of a beat. One problem arises from the overlapping of the structural elements. Under some conditions two left-hand

elliptical waves can combine to produce an apparent right-hand wave

[Pope, 196/f], Ideally, one wishes to observe the polarization of a

single element. Two possible solutions are to either select an event

in which the elements are well separated, so that there is no overlapping, or to remove the overlapping artificially with a suitable filter.

Though the author found a predominance of left-hand waves at College,

Alaska, McPherron and Ward [1965] observed that right-hand waves were

favored two to one at Flin Flon, Manitoba, also in the auroral zone.

Hessler and Heacock [1966] also observed more right-hand waves, especially when the rising tones were well separated.

Studies at low latitudes have indicated that pc 1 events sometimes

occur simultaneously at stations separated by several thousand kilometers [Tepley, 1964 ]• In contrast, Baie St. Paul and Great Whale River,

Quebec, at geomagnetic latitudes of 58.7 and 66.6 N respectively,

seldom see the same event, even though they are only separated by 700

km [Campbell and Stiltner, 1965]. Even Fort Yukon and College, Alaska,

49
only 200 km apart, show a low correlation. The implication here is

that pc l*s are fairly well localized in the auroral zone, but become

widespread at lower latitudes. Pc l's are usually seen simultaneously

at pairs of stations which are magnetically conjugate to each other

[Campbell and Stiltner, 1965; Dawson, 1965].

3. Theoretical Interpretation of Pc l f s

Most of the available evidence indicates that pc l T s are the result of a magnetic disturbance at about five or six earth radii in the

vicinity of the equatorial plane. Left-hand waves would be guided along field lines reaching the surface at auroral latitudes. Bouncing

between hemispheres would explain the regular repetition observed on

Sonagrams and also the 180 phase shift between hemispheres. The waves

could propagate to lower latitudes as a right-hand traveling in a duct

between the ionosphere and the height of maximum Alfven velocity at

about 2000 km.

The high correlation between conjugate stations at high latitudes

establishes pc l's as a field line phenomenon. The degree of correlation for conjugate stations at middle latitudes is not known. At low

latitudes travel along field lines becomes indistinguishable from propagation in the ionosphere.

The repetition times can vary from about 45 to 300 sec, though

usually it is around 180 sec [Tepley, 1965 J. It can be seen from table

6 that only L-values between 3.5 and 7 are suitable, if the repetition

time represents the time for an Alfven wave to make a round trip between

50
TABLE 6

Travel Times for Alfven Waves

Xatitude

35

v

40

u

45°

50°

55°

60°

65°

70°

L-value

1.5

1.7

2.0

2.4

3.0

4.0

5.6

8.5

Bounce time (sec)

12

15

17

23

36

69

140

390

51
hemispheres. A travel time of 180 sec would correspond to an L-value

of about 6.

As only the left-hand waves are guided along the field lines, the

bouncing waves would have to be the left-hand waves. Since these waves

cannot propagate at frequencies above the ion gyrofrequency, the equatorial value of the gyrofrequency represents the highest frequency which

can bounce on that field line. This resonance could explain why pc l*s

are never observed with frequencies higher than 5 c/s. At L = 4«5>

the equatorial gyrofrequency is 5 c/s (see fig. 8). Thus, if the bouncing

never occurs at lower L-values, higher frequencies could not propagate.

The most common type of fine-structure observed is a repeating

series of rising tones. Several workers have suggested that the rising

tones are the result of dispersion [Obayashi, 1965]. Since for a lefthand wave, higher frequencies travel slower, this could explain the

rising tones. But in order to obtain significant dispersion, — must

i

be comparable to unity, though always smaller, which would be further

evidence that high latitude field lines play the primary role. Some

pc 1 events show a fan-shaped fine-structure, i. e., successive elements

have an increasingly slow rise time. This could be interpreted as evidence of cumulative dispersion by each successive bounce. However,

only a few events show the fan-shaped structure J most events have elements with a constant rise rate. Then there is the problem of those

events which have falling tones. It seems difficult to explain all

these effects by dispersion alone. More likely the observed fine-structure is a result of generation plus dispersion.

52
53
There appears to be a qualitative difference in the pc l T s observed

at auroral latitudes and those observed closer to the equator. At College the magnetic perturbation vector was observed to be perpendicular

to the field line, indicating that the incoming wave was traveling down

the field line. At Boulder the plane of polarization was randomly oriented. Apparently Boulder sees waves arriving from a variety of directions. Effects of ground conductivity will introduce a measure of

confusion, but at least the limited polarization data available does

support the concept of left-hand waves traveling down field lines to

the auroral regions and then propagating equatorward via some other

mode. If this model is correct, waves reaching the top of the ionosphere

at high latitudes should have left-hand polarization. However, in the

transmission through the ionosphere, a right-hand wave will be generated, and because of differential attenuation and reflection, the polarization observed on the ground could conceivably have either sense.

Further evidence of the different nature of pc l*s at high and

middle latitude is obtained from studies of the areal coherence of the

events. Pc l T s appear to be a local phenomenon in the auroral zone

but widespread at lower latitudes. Again this could support the idea

of primary impingement being in the auroral zone.

There are three conceivable ways in which pc l*s could arrive at

low latitudes. If it is assumed that pc l*s reach the auroral zone

as Alfven waves, then they could propagate equatorward as an electromagnetic wave under the ionosphere. The extremely long wavelength would

put all stations in the near field of the auroral zone and pc 1 T s would

54
be expected simultaneously at all latitudes. It would also be expected

that a pc 1 event would be very widespread, when in actuality they are

not.

A more promising way to propagate pc l*s to lower latitudes is

as a right-hand wave in the duct at the top of the ionosphere. Alfven

velocity profiles for four model ionospheres are shown in figure 9»

The profiles show a rather sharp minimum at about 400 km. Thus, refractive processes could guide a right-hand wave in a duct at this altitude.

Recent measurements [Tepley, Landshoff, and Wentworth, 1965] indicate

that College sees a pc 1 six seconds earlier than Kauai, implying a

velocity of 800 km/sec, which is the right magnitude for a wave traveling at about 900 km. Presumably energy would leak through the ionosphere to be received at the ground as the right-hand wave is propagated overhead toward the equator. Apparently the wave is mostly attenuated by the time the equator is reached, as equatorial stations

see a minimum number of events. Reception of the signals from both

hemispheres could explain the double fine-structure occasionally observed

at Canton Island.

One problem is that the simple hydromagnetic theory postulated

here is not really adequate to explain this mode of propagation. About

all it can do is to hint at what might occur. In general the available

wave guide will have a height less than one wavelength, which makes

the uniform plasma assumption somewhat suspect. Collisions also become

important and can no longer be neglected. The little data so far available indicate a tendency for pc l T s to propagate along geomagnetic me55
ET TT

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56
ridians. If this is so, it will be necessary to explain why the righthand wave cannot propagate as easily east-west as it does north-south.

The answer could lie in the nature of the coupling of the two modes

in the ionosphere.

A third way in which a hydro-magnetic disturbance could propagate

to low latitudes is as a right-hand wave propagating directly in along

the equatorial plane from the original source. Unfortunately, there

are many problems associated with this explanation. If the frequency

fine-structure is the result of interhemispheric bouncing, it would

not be present at low latitudes. However, it is observed that both low

and high latitude events show similar fine-structure. Also, it would

be expected that the magnetic vector would be linearly polarized in

the north-south direction. To the best of the author 1 s knowledge, this

has not been observed. The same refractive processes that would allow

a right-hand wave to propagate in a duct under 2000 km would refract

a right-hand wave away from the earth. Only a wave coming straight

in along a radial path could penetrate the high speed region.

One possible source for pc l T s is a fast beam of protons with an

instability occurring at the Doppler-shifted ion gyrofrequency. Such

a beam could radiate hydromagnetic energy backwards at a frequency of

"iVA

CO A

where V is the beam velocity along the field line. Cornwall [1965]

showed that it was necessary for the proton velocity distribution to

be anisotropic in order for the instability to have a positive growth

rate. Fortunately, the existence of a loss cone usually provides suf57
ficient anisotropy.

Such a generative model would tend to be self-destructive. In

successive passages of the beam through the wave packet, the wave would

react on the protons so as to scatter them into the loss cone. After

a period of time the source particles would be removed and the wave

would decay, mostly because of collisional and coupling losses at the

lower elevations, bringing the whole process to a natural conclusion.

A possible objection to the foregoing is that it violates the Sturrock [1958] criterion for a convective instability. The criterion says

that only a convective instability, i. e., one that can propagate away

from its origin, is suitable for the generation of hydromagnetic waves.

In general, if the particle beam radiates in the forward direction the

instability is convective, otherwise it is nonconvective. If this criterion is applicable, a proton beam can only produce a right-hand wave

and an electron beam a left-hand wave.

Stefant [1965] and Gendrin [1965] showed that conditions for the

transfer of energy from a proton beam to a hydromagnetic wave were particularly favorable when the frequency was either near half the electron gyrofrequency or twice the ion gyrofrequency. At these frequencies the group velocity of the wave is equal to the beam velocity and

the transfer of energy may be made over a large distance. Gendrin also

showed that the frequency range over which the instability can take

place is greatly enlarged under these conditions.

As mentioned earlier the proton beam would feed energy into a righthand wave, but it is desirable to have a left-hand wave which will be

5S
field guided and thus can penetrate the ionosphere. Gendrin suggested

the possibility that a beam could interact simultaneously with both

hydro-magnetic modes and could serve as a means to transfer energy from

one mode to the other, but he did not outline explicitly the conditions

under which this could happen.

Stefant considered the case when the beam velocity is slightly

different from the group velocity. This produces two right-hand waves

at about twice the ion gyrofrequency, which could beat together to produce a left-hand wave at a much lower frequency. If a nonlinearity

exists, an independent left-hand pc 1 wave could be generated, which

could penetrate the ionosphere, where the higher frequency carriers

would be greatly attenuated. The beating also serves the purpose of

producing a frequency low enough to explain pc 1 occurrence.

So far the discussion has centered about a model which relies on

a particle instability mechanism to generate hydromagnetic waves which

are observed on the ground as geomagnetic micropulsations . Other speculations have involved the idea of hydromagnetic waves generated by

fluctuations in the magnetopause.

Typical of such models is the one suggested by Jacobs and Watanabe

[1962]. The interaction of the solar wind with the magnetopause is

postulated to generate noisy fluctuations which propagate as left-hand

waves along the geomagnetic field lines down to the ionosphere at high

latitudes. As the original fluctuations are noisy, some sort of filtering mechanism must operate so that only regular pulsations are transmitted to the surface. Jacobs and Watanabe suggested that standing waves

59
are established between the top of the ionosphere and the height of

maximum Alfven velocity. (Others have suggested that the entire field

exhibits resonances, which are observed as regular pulsations. Except

for field lines at very low latitudes, the fundamental resonance is

much too long in period to account for pc lT s and there is no reason

to expect to see only the higher harmonics.) The effective width of

the transmitted frequency determines the envelope or the beating of

the pc l T s. Secondary waves are then generated which propagate to lower

latitudes via the mechanism discussed earlier.

This model can explain the characteristic frequency, the beating,

and the polarization of pc l T s, but it fails to explain the opposite

phases observed between conjugate points and the series of rising tones

seen on Sonagrams. In a second paper Jacobs and Watanabe [1963] changed

their model to one of bouncing particles in order to remove these deficiencies.

In the revised model bouncing protons, instead of stochastic fluctuations, generate standing waves in the aforementioned duct. To account for the observed fine structure, protons are believed to be bouncing

at several latitudes simultaneously. Since the Alfven velocity and

the angle of field line inclination is greater at higher latitudes,

emissions stimulated at high latitudes will have a higher frequency

than those generated at middle latitudes. At the same time, protons

bouncing at the higher latitudes will have a longer bounce period, and

therefore after each successive bounce, the higher frequencies will

be further retarded with respect to the lower frequencies, producing

the fan-shaped structure.

60
The difficulty with Jacobs and Watanabe T s revised theory is that

the predicted resonant frequency is greatly dependent on local ionospheric

conditions, which will often differ drastically between conjugate points.

In contrast, simultaneous Sonagrams taken at conjugate points show a

similar frequency structure. Also, the fan-shaped structure is not the

most common type, and the theory cannot explain the other forms observed.

Wentworth and Tepley [1962] postulated another type of bouncing

particle model, one using electrons wnose bounce period is equal to

the wave period. The diamagnetic effect of the spiraling electrons

would generate a hydro-magnetic wave which is detected on the ground.

Two fundamental difficulties are that beats are predicted to occur simultaneously in both hemispheres and the magnetic polarization should

be aligned along the field lines. The reason for the latter point is

that spiraling electrons would produce a perturbation magnetic field

aligned against the main field. Since k . b = 0, this could only produce a right-hand wave propagating across the field lines. Wentworth

later rejected the bouncing electron model in favor of a proton model,

whose bounce period was equal to the repetition time. Now the 180

phase shift was accounted for. A summary of all the particles was given

by Tepley [1964]. As pc lf s can last for several hours, all these models share the formidable problem of explaining the stability of the

postulated particle bunches. Finally, no good correlation between pc 1

events and particle events has been shown.

Models for the generation of pc lf s fall into three general categories, hydromagnetic, particle, and particle-wave interactions. All

61
of them have serious faults, but overall, the particle-wave models seem

the most satisfactory. The question of whether an instability must

be convective in order to serve as a satisfactory model needs to be

settled. An acceptable theory must explain the conjugacy, the observed

fine-structure, its hemispheric phase shift, the long duration of the

pulsations, and the lack of correlation with other geophysical phenomena.

Acknowledgments

A portion of this work was done at the Geophysical Institute of the

University of Alaska under contract number NONR 3010 (02). Much helpful

criticism was supplied by M. Sugiura of Goddard Space Flight Center.

I also wish to thank T. Watanabe of the University of British Columbia,

J. Cornwall of the Aerospace Corporation, R. C. Wentworth and L. Tepley

of the Lockheed Missiles and Space Company, F. X. Bostick of the University of Texas, and R. Stefant of the Institute for Telecommunication

Sciences and Aeronomy for helpful discussions. This research was supported by the Advanced Research Projects Agency under contract number

183.

62
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