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CONFERENCE ON NON-LINEAR

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DECEMBER 16-17, 1963

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technical *^oU 211, Volume, M

Issued April 22, 1964

CONFERENCE ON NONLINEAR PROCESSES

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Volinne IV

Table of Contents

Session 3- Collisional Radio-Wave Interactions. Ifeirt II

Chairman: M. Cutolo

3.1 Microscopic theory of ionospheric cross -modulation.

David Layzer and Donald H. Menzel, Harvard College

Observatory 1

3.2 Ionospheric wave interaction using gyro-waves. Part 1.

The night-time lower E-region^ R. A. Smith,, The University

of New England, Armidale, N.S.W. , Australia 39

3.3 Wave interaction at oblique incidence, near the gyro

frequency. F. H. Hibberd, Ionosphere Research Laboratory, The Pennsylvania State University, and University

of New England, Annidale, N. S.W. , Australia 45

3.4 Non-linear propagation of electromagnetic waves in

magnetoplasmas. Mahendra S. Sodha and Carl J. P^limibo,

Republic Aviation Corporation, Parmingdale, N. Y 6 5

3.5 Collision effects in hydromagneto- ionic theory, Hari K. Sen, Air Force Cambridge Research Laboratories, and Arne

A. Vfyller, Williamson Development Co., Inc., Concord,

Mass 73

3.6 A solution method for cold plasma, R. E. Hartle and

J. J. Gibbons, Ionosphere Research Laboratory, The

Itennsylvania State University 117

111

Ionospheric Cross -Modulation: A Microscopic Theory

David. Layzer and Donald H. Menzel

Harvard College Observatory

In 1937 Bailey and Martyn proposed a theory of ionospheric cross-modulation along the follov?ing lines. Radiation emitted by the disturbing transmitter is strongly

absorbed in a region of the ionosphere where the wanted

wave is refracted and attenuated. Absorption of the disturbing radiation raises the temperature of the ionosphere.

If the amplitude of the disturbing radiation varies with

time, so will the resulting temperature increase and hence

the attenuation suffered by the wanted wave. In the simplest

experimental arrangement the wanted wave is initially

unmodulated and the disturbing radiation is modulated at

a definite audio frequency. In traversing the ionosphere,

the wanted wave becomes modulated at the frequency of the

disturbing radiation and also at twice this frequency.

The Bailey-Martyn theory maXes three quantitative

predictions concerning the transferred modulation. The

first concerns its dependence on the power of the disturbing radiation, and involves only the assumption that the

effect is small enough to warrant a linearized description.

The remaining two predictions concern the dependence of the

transferred modulation and its phase lag on the modulation

frequency. These predictions involve the additional
assumption of a unique relation, valid for all values

of the modulation frequencyj between the mean thermal energy

and the mean collision frequency of the electrons. A relation

of this kind can exist, however, only for modulation frequencies much less than 150 C/s. Since the experimental range

of modulation frequencies extends to nearly 1500 C/s, the

foundations of the Bailey-Martyn theory need to be reconsidered.

This paper describes a microscopic theory of ionospheric

cross-modulation. The velocity-distribution function for

the electrons enters explicitly into the theory; it is

determined by a differential equation whose form depends

on the modulation frequency. Knowing the distribution

function, one can calculate the absorption coefficient.

The form of the predicted absorption coefficient depends

on the assumed form of the electron-molecule interaction

law. By numerical methods, we have calculated the transferred modulation and its phase lag as functions of modulation frequency for a few simple interaction laws. The

calculations show that the effects of departures from a

Maxwellian velocity distribution are indeed significant.

The predictions are sensitive to the assumed form of the

electron-molecule collision law. Although the present theory

is still highly idealized, the results obtained suggest
that further theoretical and experimental refinements could

lead to an experimental determination of the electron-molecule

interaction law in the D region.

1. The Bailey-Martyn Theory

The pa.'ssage of a radio v;ave through a region of the

ionosphere in which it is partially absorbed raises the

electron temperature in the region and hence changes its

absorption coefficient. The absorption process is accordingly non-linear: the attenuation suffered by each of

several radio waves passing through the same region depends

on that suffered by all the others. This is the physical

basis of Bailey and Martyn's [1934] theory of ionospheric

cross-modulation

In a typical cross-modulation experiment an initially

unmodulated carrier wave (the "wanted" wave) traverses a

region in which a second wave at a different carrier

frequency (the "disturbing" wave) is heavily absorbed.

The disturbing wave is amplitude-modulated at a definite

frequency. On reception, the wanted wave is found to be

modulated at the same frequency and also at twice this

frequency.

The main predictions of the Bailey-Martyn theory do

not depend on a detailed description of the physical processes involved^but follow from a few general assumptions,

the first of which is that the phenomenon admits a linearized

description. The linearized energy-balance equation for

3
the electron gas has the form

1^ = KI - T-^€ . (1.1)

Here e denotes the difference

€ = Ik(T - T ) (1.2)

2 e m

between the mean thermal energy of the electrons and the

mean thermal energy of the molecules^ k denotes the absorption coefficient of the medium, I^ denotes the intensity

of the ambient radiation field, and t denotes the thermal

relaxation time of the electron gas. As (1.1) is linear,

we may treat the individual Fourier components of the

radiation field separately. The oD-component of (1.1) is

ioe = Kl - T-^e , (1.3)

O) O) CD

which has the solution

Equation (1.1) has the same form as that governing the

current in an RL circuit; x represents the time constant

of the circuit and I^ the applied e.m.f

,

-icp KTI KTe I

/(l+T'^ao )
The angle cp represents the phase difference between the

oD-component of the radiation field and the co-component of

the electron temperature. In the D and lower E regions,

T ^ 100 secj so that the attenuation factor (1+t^od^) "^/^

is very small except for audio frequencies.

The assumption of linearity also implies that the

variable part of the electron-molecule collision frequency

V oc e. According to Lorentz's theory, k « v. Hence, setting

K = Kq + / K e'^'^ do) , V = V5 + / V e dco , (1.5)

we have

K « V oc € . (1.6)

a> o) CO

Finally, if the region where appreciable modulation transfer

occurs is nearly homogeneous, we have

X oc K , (1.7)

00 O)

where X describes the depth and phase of the transferred

modulation. From (1.4), (1.6), (1.7), we have

X = I T e~^^ ; T = (1 + T^GO^)-^/^ , (f = tan~^(cDT) . (1.8)

Oi O) CD O) CO

The predicted linear dependence of X on I , the predicted variation of the phase lag cp with 00, and the
piredicted variation of the coefficient T v;ith od can all

be tested separately,

(i) Linear dependence of X on I . If the disturbing

wave is modulated to a depth M at a single frequency co and

if P denotes the total pov/er of the disturbing radiation,

then

I « P[ (1 + ^if) -!- 2MCOSC0t + iM^GOS2C0t] . (lo9)

Ratcliffe and Shaw [1948] verified that X « P over a wide

CD range of P. Huxley et al.[l947, 1948] verified the predicted

linear dependence of P on M and showed that the predicted

quadratic dependence on X on M was qualitatively in agreement with experiment. These experimental results show that

a linearized description of ionospheric cross-modulation is

valid over a wide range of experimental conditions .

(ii) The phase lag cp (cd) . Ratcliffe and Shaw [1948] and

subsequent workers found that they could secure good agreement between predicted and measured phase lags over a wide

range of co by choosing the parameter x appropriately. The

required values of t seen to be consistent with values derived

by extrapolating laboratory measurements [Huxley, 1959]

(iii) The coefficient T . Formula (1.8) for T agrees

CD Oi qualitatively, but not quantitatively, with the experiment
[Ratcliffe and Shav, 1948; Huxley> 1950]. The discrepancies

between theory and experiment, though not large, appear to

be significant. Huxley [1950] suggested that the two spatially

separated regions in which cross-modulation occurs in a typical

experiment — one on the ascending, the other on the descending

branch of the wanted ray — may have sufficiently different

properties to invalidate the assumption of homogeneity underlying (1.6) . Calculations based on a two-center model [Huxley,

1950] do not, however, significantly reduce the discrepancies.

The derivation of formula (1.8) actually rests not only

on the assximptions of linearity and homogeneity but also on

a third major assximption: that, at any given point in the

ionosphere, the quantities k, v, and t depend only on the

electron temperature and not explicitly on the modulation

frequency. Now, the relation between collision frequency

(say) and electron temperature depends in general on the

form of the velocity distribution of the electrons. If the

distribution is Maxwellian, or has any other fixed form

depending on a single parameter, it is completely specified

by T , so that v becomes a function of T . But if the * e e electron temperature does not serve to specify the distribution completely, the relation between v and T may be

many-valued. In the present problem, the variable part
of the distribution changes appreciably in a time of order

(xT^ . Since this is short compared with the thermal relaxation

.time

't,there is no reason to suppose that the distribution remains

accurately Maxwellian. It is true that if the intensity of

the incident radiation field is sufficiently small, the

velocity distribution of the electrons will be approximately

Maxwellian. But the phenomenon of cross-modulation depends

entirely on the variable component of the velocity distribution^ and, as we shall see, this component remains nonMaxwellian even in the limit of vanishing field intensity.

This paper presents a theory of cross-modulation in

which the electronic velocity-distribution function figures

explicitly. Although the present theory is more realistic

than the macroscopic theory sketched above, it is oversimplified in one important respect: It treats collisions

between electrons and molecules as if they were perfectly

elastic. In reality, cooling of the electron gas in the

D and E regions results chiefly from the collisional excitation of molecular nitrogen. Caldirola and De Barbieri

[1963] have recently succeeded in extending the present

theory to allow, at least approximately, for inelastic

collisions.

8
For the sake of simplicity, the following discussion

ignores the effects of the earth's magnetic field. This

is permissible only if all the carrier frequencies that

figure in the discussion are much greater than the gyrofrequency. The modifications required v/hen this condition

is not met are straightforward; they are described in the

paper by Caldirola and De Barbieri mentioned in the last

paragraph.
2. Reduction of the Boltzmann Equation

The Boltzmann equation for an electron gas in an electromagnetic field has the form

rr + q'T- + - E + - X B )'^— = TT . (2.1) ot ~ ox m \ ~ c -v y oq ot

The follov^ing discussion of this equation is a generalization

of the treatment of a weakly ionized gas in a constant electric

field given by Chapman and Cowling [1939, pp. 346-352].

We assume at first that the field is that of a linearly

2 polarxzed plane wave propagating xn the z-directxon:

2 We adopt the convention that when a real quantity, such as

E or f, is represented by a complex express ion « as in (2.2)

and (2.6), the real part of this expression is to be under-

(2) stood. Note, however, that the quantities E, F, k, f ,

etc., are all complex, and that both the real and imaginary

parts of an equation like (2.11) are significant.

E = E e^(^^2-P^) , B = Sk E e^(k2-pt)

j X y p

The distribution function f depends on the six variables

Zt <l> t. since |ck/p| % 1, the magnetic force is of order

q/c compared with the electric force. Similarly, the second

10
term on the left side of (2.1) is of order q/c compared with

the first term. Neglecting terms of this order, we obtain in

place of (2.1)

bf „ df 6f ,^ ,, aT-^^x^^aT' ^2.3)

where

F = - E . (2.4)

'\* m 'v.

In this approximation, f depends on only four arguments:

f = f(z;u,q;t) . (2.5)

We may expand f in Legendre polynomials of the argument

u/q. We shall need only the first two terms in the expansion:

f = f<°Nq,t) + iPe^<^='-P^^f<^Nq,t) . (2.6)

If the number density n of electrons is sufficiently

small compared with the number density of molecules, we may

neglect electrons-electron and electron-ion collisions. The

collision term in the Boltzmann equation, represented by

6f/6t, is then a linear functional of f:

|f=A,f) .

Lorentz showed that

11
(t) A[ug(q)] = - v(q)ug(q) ( S ) > (2.8)

where ro/M is the electron-molecule mass ratl9> v ie the

velocity*-depehd^t collision frequency, and g is an arbitrary function, Davydov [1935] derived the important

formula

By inserting (2.6) into (2.3) and using the formulae of

Lorentz and Davydov to evaluate the right side of the resulting equation > we obtain differential equations for the

functions f^^' and f^^^.^

3 (k) The differential equation for f , the coefficient of

P, (u/q) in the Legendre expansion of f, involves only the

functions f

-^ with j < k, so that the sequence of approximations may be terminated at any point. By contrast, the

equations for the moments of f of order k involve moments

of order k + 1.

The equation for f^ ' is

Mf^^' (p + iv) + 2|_— « (2.10)

12
or

p + iv q dq

The left side of the equation for f has the form

at (-X It ( ^v^'" )) • <=-i='

Here the brackets indicate a double average: over direction

in velocity space (to eliminate terms containing Legendre

polynomials P. (u/q) with k > 0) , and over time (to eliminate

radio-frequency fluctuations in f , which are of no physical

interest) . Since we need to preserve audio-frequency variations of f , the time- aver aging must be over an interval

that is short ccanpared with the reciprocal of the highest

modulation frequency of interest as well as long compared

with the reciprocal of the carrier frequency p; thus we must

have p » o). Application of the well-known rule

<Re Ae^^^-Re Be^^*>^ = j Re AB , (2.13)

where A denotes the complex conjugate of A, gives

/f f ('iPuf(^>>

-i-'^iKr^)'^!: (t^)}

13
(0) .,2 :, / , :..(0)

Since

- 2 p2 + v= I*"! Lq Sq + q a<3 ^, q Sq jj ' <2''*'

"^-^[M-(?)]'

averaging over direction in velocity space gives

<^.^(-x-<^')>

(0) , :, / , :,.(0)

-2p- + v2i^i Lq'-^r'* 3^^U"5r";J

6 p^ + V*

Combining this result with (2.9) we hav^fc, finally

St

where

-S^^{^^v<^>[^""*|^]}-0' <-"'

e = r" !

' . -^ —^ > (2.17)

6 m p2 + V2 m

We may immediately generalize these results to an

arbitrary radiation field. The electric vector at a given

point may be written in the form

E = y E. , (2.18)

•v = ISj '

u
where each Fourier component E. represents the electric

field of a lineairly polarized plane wave. In place of (2.6)

we have

f = f^°^(q,t) + y iF. (k.-q)f]^Nq,t)exp(ik.-x - p t) (2.19)

and in place of (2.10)^

ff^> =--4-T-¥^. (2.20)

D p^ + IV q dq

Finally^ the function f ^ satisfies (2.16) with 6 given by

® = r- ) 2I .B + -^

(2.21)

6 m £_, p2 + v2 m

instead of by (2.17).

15
3. The Dispersion Relation

The electric current density produced by the radiation

field (2.2) is given by

= iSFcxjfi<Oh(^-^^) , (3.1) V3p+ivy'

jy = j^ = . (3.r)

From Maxwell's equation,

1 . curl B = - E + 47rj , (3.2)

and (3.1)^ we obtain

^ mj \3p+iv

P ' m J ^ ^ V 3 p2 + v^

^ i4iel r,(0)^/^^:^N ^ m J \ 3 p2 + v2 y

16
In order to write this relation in a more compact form we

introduce the plasma frequency po, defined by

P? = ^> (3.4)

and use the abbreviation

<*>=i /f'°'d [f q=*(q)] , (3.5)

where n denotes the electron density. The dispersion

relation (3.3) then takes the form

(ck)^ = p^ - P§( pa l\^y + iPo\;FT-^/ • <3-^)

The absorption coefficient k is defined by

k = kc + JiK , (3.7)

where ko and k are both real. The following approximate

formulae^ valid under the conditions stated, are often

convenient

Po / pv \

"=*" = 7" \p2 + vg /

<P° « P> ' ^^'^^

CK =

(^ ~ j

<v > (po « p, V « p) . (3.9)

The results of this section were first obtained by

Lorentz

17
4. The Absorption of Unmodulated Radio Waves

When o) = Oj (2.16) reduces to an ordinary first-order

differential equation, whose solution is

f = nC exp

q

/ e"^qdq

where is given by (2.21) and the constant C is determined

by the normalization condition

(0)_2

(4.1)

47r / f ' 'q^dq = n . (4.2)

Formula (4.1) is valid for all values of the carrier frequencies p. occuring in formula (2.21) for 0. We recall that

f represents the isotropic part of the distribution function

averaged over a time interval long compared with all the

characteristic times pT^ . If the radiation field has only

a single Fourier component, f represents the isotropic part

of the distribution function averaged over a single period.

Formula (4.1) remains valid in the limit p = (E = const.).

In this case

1 M iFl^ ^^,m

and (4,1) coincides with a formula derived by Chapman and

18
Cowling [i960; eq. (13), p. 350].

When the carrier frequency is very large, 6 is given by

8 = ^ !

-^ + -;f (P » V)

, (4.4)

f is accordingly Maxwellian, the electron temperature being

given by Lorentz's formula,

1 mIfI^

kT = kT + ^

' . (4.5) e m 6 p2

If the molecules are Maxwellian (collision frequency independent)

of velocity) , the distribution function is Maxwellian for all

values of the carrier frequency. Otherwise the distribution

function departs markedly from the Maxwellian form when

G » kT /m and p ~ v. m

If the molecules are rigid elastic spheres (v = q/l,

I s const.) the integral in (4.1) may be evaluated explicitly;

one obtains

f^°^ = C'(i^ + i^ + A)^ e"^^ (4.6)

where

"^ " 2kT ' P ~ 2kT * "^ " 2kT • ^^''^ m m m

19
5, The Variable Part of f

In this section we derive the equations governing small

departures from the steady-state velocity distributions discussed in the preceding section.

Let

e(q,t) = eo(q)[l + 0(q,t)] . (5.1)

The function ©o coincides with the time-independent function

that was called G in the preceding section. We define new

dimensionless variables x, X, cp:

q

qdq 1

<3^ P (n\

<i3c = -— , X = 7 ZT ' <P(x,t) = 47rn-^ q^f ^''^ (q,t)dq . (5.2)

>•

Note that

2 Oo

dx ^ dX d0o

^ (5.3) X X Go

The function cp is the probability distribution function of

the dimensionless velocity variable x;

<p(a,t) = Pr{x « a} . (5.4)

It follows that

9(0, t) = , cp(«',t) = 1 . (5.5)

The partial derivative Scp/Sx is the probability density

associated with x. In terms of the new variables (2.16)

20
takes the form

I? = f4-.|r(«-^).(x-|)||

*«[-|l^*'^<,|^(«o^ll)]}. (5.6,

So far v/e have not made any assumptions about the magnitude of the function ©(q^t). In order to reduce (5.6) to an

ordinary differential equation we now set

<P = <Po(x) + cpi(x,t) (5.7)

and assume that and <Pi are so small that their product may

be neglected in (5.6), This will be true if at least one of

the conditions ~ kT /m, M « 1, is satisfied. Having linearized (5,6), we can deal separately with the Fourier components

of th© radiation field, just as in the elementary theory of

Section 1, Let

e(x,t) = e^(x)e^'"^ , cpi(x,t) = <p^(x)e^"'^ , (5.8)

It is convenient to replace the modulation frequency cd by

a dimensionless variable and to separate out the velocity

dependence of the function V (x) . We accordingly define

'^^''^ = VOO ' ^ " J

v(x)d(p^ , (5.9)

21
^ = 2(m/M)v • (5-1°^

Finally^ since cr^ satisfies the equation

we have

= Ceysx^/^e"'' , (5.12)

where C is determined by the normalization condition cp^, (») = 1

Hence

- X

^ = - C(e^X)V^e"' . (5.13)

Omitting second-order terms in (5.6) and using (5.9), (5.10), and

(5.13), we obtain

^'oTA'-.''^)*(--i)^^ ^«^'. = «.CeV^x3Ae-. (5.14,

since cp and cpo both satisfy the boundary conditions (5.5),

<p must satisfy the boundary conditions

cp (0) = cp (oo) = . (5.15)

CD CO

Equations (5.14) and (5.15) together with the dispersion

relation (3.6) represent the formal solution of our problem.

The functions Gq and 6 are defined by the radiation field.

22
Given @o, one finds the function 3t(q) and X(q) from (5.2).

Equation (5.14) with the boundary conditions (5.15) may be

integrated numerically by the method described in the Appendix.

Finally, knowing cp ,one can calculate the variable part of the

absorption coefficient from the dispersion relation.

Equation (5.14) assumes a simpler form when 6 ?« kT /m,

m

so that the unperturbed distribution function is nearly

Maxwellian. We may then write

X = X , )o = const. , 6 = const. , 00 (5.16)

so that (5.12) becomes

hx ~ Vir

Setting

S<Po 2 1/2 -X "^— = T" X ' e (5.17)

cp (x) =

CO - -7- y(x) , VTT or (5.18)

we obtain in place of (5.14)

xy" +( X-— jy' - iar\Y = - x^'^e

The boundary conditions are

(5.19)

y(0) = y(oo) =0 .

In the numerical work described in the next section the

function t] (x) is taken to have the for.n

/ X 2(r + ^)t -r

(5.20)

(5.21)

23
6. Some Numerical Results

Miss Cara Joy Hughes employed the IBM 7094 computer

at the Harvard University Computing Center to integrate (5.19)

with the boundary conditions (5.20) and with t) (x) given by

(5*21) . Integrations were carried out for every pair of

parameter values (a,r) with a in the set (.1, .2, .5, 1, 2,

5, 10, 15) and r in the set (-.5, .5, 1, 1.5). The integration

procedure is described in the Appendix,

For r = the solution of (5.19) is

3/2 -X

y = rr-i^ (r = 0) . (6.1)

The complete distribution function f corresponding to this

solution is Maxwellian with a variable electron temperature.

The factor (1 + ia)~^ in (6.1) corresponds to the factor

(1 + ia)T)-^ in (1.4).

Figures 1 and 2 show the modulus and argument of the

function

y = (1 + ia)y (6.2)

for r = -.5, 1.5 and for a = 2, 5. In both figures the

curves corresponding to r = -.5 and r = 1.5 differ markedly

from the curve for the Maxwellian case r = 0. For r = -.5

the peak of the function ly(x) | occurs at a smaller value

2k
of X than for r = 0, while for r = 1,5 it occurs at a larger

value. In the four cases with r f^ the function arg y changes

sign near x = 1.5. For large positive values of x, arg y is

positive for r = 1.5, negative for r = -.5.

According to (3.9), the absorption coefficient k varies

directly as the mean collision frequency (the mean being defined

by (3.5)) if the carrier frequency is sufficiently high. By

(3.5), (5.2), (5.7), (5.8), (5.18), and (5.21),

00 00 « - (2r + 3)r j cp^x^^^dx cc - (2r + 3)re j yx^-^dx . (6.3)

Just as in Section 1, the assumptions of linearity and homogeneity imply that X « k (see (1.7)). Hence

00 X * (2r + 3)re / yx^-^dx . (6.4)

O) 00 J

*

Note that X vanishes for Maxwellian molecules (r = 0) . This

CO means that the elementary theory becomes rigorously valid in

the limit when the phenomenon it describes disappears.

By analogy with (1.8), we write

00Tyx^-Mx = T(a)e"^^^°'^ . (6.5)

25
The functions cp(a) and T(a) are shown in Figures 3 and 4

for r = -.5, 0, ,5, 1, 1.5. For r ^ the curves cp(a) have

roughly the same shape. Compressing the horizontal scale of

the curve for r = .5 by 25 per cent would bring it into nearcoiAcidence with the curve for r = 0, but for r = 1, 1,5, the

scaling factor increases markedly with x. For r = -,5 (the

law appropriate to collisions between electrons and ions )

,

the phase lag at first increases more rapidly with increasing

modulation frequency than in the Maxwellian case, then more

slowly. The shapes of the curves for r = -.5 and r = are

entirely different.

If one were to use the phase-lag curve for r = to

analyze experimental results relating to a hypothetical gas

in which the electron-molecule collisions were elastic and

were characterized by a velocity-independent free path (the

case r = .5), one would over-estimate the mean collision

frequency v by 25 per cent. If a higher value of r were

appropriate, the error could be much greater.

Turning now to the coefficient of transferred modulation T(a) (Fig. 4), we see that the curvature of the

function T(a) increases with increasing r over the entire

range -.5 < r < 1.5. Compressing the horizontal scale of

the curve for r = ,5 by 25 per cent would make it fall off

I

26
more steeply at small and moderate values of a than the curve

for r = 0. In general, one cannot devise a horizontal scale

transformation that, for a given value of r ?^ 0, will make

both T(a) and <p(a) assume the forms appropriate to the case

r = 0.

To sum up, the numerical calculations show that significant departures from the predictions of the elementary theory

may be expected. Moreover, values of the mean collision

frequency derived by using the elementary theory to analyze

cross-modulation data may be significantly in error. On the

positive side, the sensitivity of the predictions to changes

in the collision law suggests that further theoretical and

experimental refinements could ultimately lead to em accurate

experimental determination not only of the collision frequency

but of the form of the collision law.

Much of the material in this paper is described in an

unpublished report by the present authors dated March 14, 1959,

and supported by the U. S, Air Force.

27
Appendix

Cara Joy Hughes

Division of Engineering and Applied Physics

Harvard university

The following paragraphs describe the

method used to integrate (5.19). The same

method can be applied to the more general

equation (5.14).

28
Consider the inhomogeneous second-order linear differential equation

a(x)y" + b(x)y' + c(x)y = d(x) . (A.l)

We may approximate (A.l) by the set of coupled difference

equations

a.A^y. + b.Ay. + c.y. = d. (i = l,...,n-l) (A. 2)

2. t l"*! XX X

where

a.XX= a(x. ) , etc. (A. 3)

x. = Xo + ih (A. 4)

Ay^ ^

. A^y. . (A.5)

(A. 2) is a set of (n-1) equations for the (n+1) variables

yo*yiJ»«»*y • The two boundary conditions provide two additional equations, so that in general the set of difference

equations together with the boundary conditions have a unique

solution

In the problem at hand, yo and y are given:

yo = Yjj = . (A, 6)

The equations (A. 2) thus have the form

29
/ aa bg \ / 2ai \

2a:

=: d. (2 < i ^ n-2)

1

2an- " -g n-g

" 2h y = d

•'n-g n-x

(A. 7)

In matrix notation.

MY = D (A. 8)

v/here M is a tridiagonal square matrix and Y«D are column

matrices.

To solve (A. 8), we write M in the form

M = LR { ->, r "i*

where L and R have the fojnms

L =

1 0. .aO

la 1 0.4 *0

la 1.4 .0

11

n

\\

ki ri .

1<;8 rz . R = ka rs

• • • •

. . .

..0

..0

..0

..k

n

The coefficients l.,k.,r. must satisfy the equations

Mil = kijMis = ri

(A.IO)

M. . 1

. , = l.k. , M. . = l.r. +k., M. . = r. (2 < i < n- ^x-x 1 i_i' 1^1 ''x i_.i i' 1,1+1 i

^ 1)

M^«i=^k ,M =lr +k n,n-x n n-x' n,n n n-i n

(A. 11)

30
(i^t)-^ (-%-0- = ^^ (^^i)^-- (-^-i)yi^ aj-i _ t>i-i

h2 2h yi-i

= d; (2 < i < n-2)

(-^^--0^"-^( an-a _^ bn-a

h2 2h

yn-2 = ^n-:

In matrix notation

(A. 7)

MY = D (A. 8)

where M is a tridiagonal square matrix of ord^r (n-l)^ and

Y>D are column matrices.

To solve (A. 8) 4 we write M in the form

M = LR (A. 9)

where L and R have the forms

L = 10

la 1

I3 1

• • • • ln-1 1

R =

ki ri . . . ka ra . . . ks ra . .

^n-i

The coefficients I., k., r. must satisfy the equations

(A. 10)

mi,i = ki , nii,a = ^1

mi,i-i = lil«i-i * mi,i = Uri-i+ki , mi,i+i = ri (2 « i ^ n-1)

(A. 11)

31
which may clearly be solved in serial order for ki, ri. Is,

ka, rsj etc. Thus L and R are completely specified by M.

Let

RY = U .

Then (A. 8) becomes

LU = D

or, again equating matrix elements.

ui = d,

liui_i + ui = di (2 :g 1 ^ n-1)

which can be solved serially for the u.. Having found U,

we then determine Y from (A. 12), giving

l^iYi + riyi+i = u; (1 n-2)

(A. 12)

(A. 13)

(A. 13')

(A. 12')

^n-iYn-i - ^n-i

which can be solved in reverse serial order, beginning with

the last equation in the set. The solution is now complete.

One can estimate the accuracy of the procedure described,

as applied to the differential equation (5.19), by comparing

numerical solutions of this equation for t] = 1 with the exact

solutions (6.1). The numerical solutions were carried out

for a grid spacing h = 0.05 and with the approximate boundary

condition y(x = 25) = in place of y (x = oo) =0.

32
References

Bailey^ V, A,, Phil. Mag 23, 929 (1937).

Bailey, V. A,, and D. F. Martyn, Phil. Mag. 18, 369 (1934).

Caldirola, P., and O. DeBarbieri, on some non-linear phenomenon

in the ionospheric plasma. Radio Science (1964)

Chapman, S., and T. G. Cowling, The mathematical theory of

non-uniform gases, second ed. (Cambridge University Press, 1952)

Davydov, Phys, Zeit Sowjetunion 8, 59 (1935).

Huxley, L. G. H., Proc. Roy. Soc. A, 200 , 486 (1950).

, Nuovo Cim.Suppl, (Series 9) £, 1 (1952).

, J. Atmos. Terr. Phys. 3j6, 46 (1959).

Huxley, L. G. H., Foster, H. G., and C. C. Newton, Nature 159 ,300 (1947)

, Proc, Phys. Soc. 61,

13^ (1948)

Ratcliffe, J. A., and I, J. Shaw, Proc. Roy. Soc. A, 193, 311 (1948),

For a complete bibliography up to 1952 see Huxley (1952)

33
Legends

Fig. 1 The function jy] (see Eqs. 6,2, 5.18) for representative

values of r and a

Fig. 2 The function arg y (see Eqs. 6.2, 5.18) for representative values of r and a

Fig. 3 Phase lag of the transferred modulation (see Eg. 6.5)

as a function of modulation frequency for various

collision laws

Fig. 4 Amplitude of the transferred modulation (see Eq. 6.5)

as a function of modulation frequency for various

collision laws

1

34
lyl

.6

ir; Fig. 1. The function |y|(see Eqs. 6.2, 5.18) for representat

values of t and a.

ive

argy

Fig. 2 The function arg y (see Eqs. 6.2, 5.18) for representative values

of r and a

35
(0nl

in

9 vO

U ^

i) ^ ^

to flo

o

m

I—

1

o

tK n1

<u >

^1M h

i)

M-t VH

CO

4J

o

n

(U rr

.C (1) !-> M

GO O "I

0, a

m

04)

36
37

Wave Interaction Research at the University of New England

R. A, Smith

University of New England

New South Wales, Australia

At the University of New England, Armidale, New South Wales,

Australia, investigations of radio wave interaction in the ionosphere

have been characterised by the use of extraordinary gyro-waves

'

to enhance the electron heating and by progressive increases of the

power flux to levels for which the electron temperature has greatly

exceeded the temperature of the gas. Because of the different naethods

employed to study this heating, the research can be divided into two

periods. From 1950 to 1957 the experimental conditions were chosen

to conform to the requirements of the theory of gyro -interaction

given by Bailey in 1937 and 1938, the disturbing wave being a pulsed

gyro -wave transmitted vertically and the wanted wave a low frequency

wave which passed at an oblique angle through the region of the

ionosphere acted on by the gyro -wave. Since 1959 the wanted wave

has also been pulsed and transmitted vertically, the techniques due

to Fejer (1955) being used to measure its interaction with the pulsed

gyro -wave. The latter experiments have been made both at night

and during the day and have added considerably to knowledge of the

fundamental processes occurring in radio wave interaction and of the

properties of the regions of the ionosphere where this interaction

occurs. The principal experimental parameters in each period

and the results obtained by the two methods are now summarized.

1950 - 1957

The experiments (Bailey et al. , 1951; Smith, 1957) were carried

out between 1950 and 1954 in collaboration with a group from the

University of Sydney under Professor V. A. Bailey. The disturbing

transmitter was located at armidale and generated pulses of 1 ms

duration and 36 kw power at a number of frequencies within +_ 20% of

the gyro -frequency (1. 515 Mc/s). Its antenna was a horizontal halfwave dipole suspended between two 150 ft. self-supporting towers.

*The term gyro-wave is here used in the sense of a wave whose

frequency is sufficiently close to the gyro -frequency for the absorption

of the extraordinary wave mode to be significantly higher than that of

the ordinary wave mode.

39
The wanted wave (0. 59 Mc/s), transmitted as an unmodulated carrier

wave by the National broadcasting station 4QR, Brisbane, Queensland,

was received at Katoomba, New South Wales, 740 km from Brisbane

(Armidale is close to the mid-point of a great circle path between

Brisbane and Katoomba). The envelope of the impressed modulation

pulse was displayed on an oscillograph and photographed at 10 -second

intervals. Observations were rejected when there was any evidence of

selective fading.

The plots of impressed modulation against frequency of the disturbing wave were found to be of resonance form with two peaks

separated by a central minimum at the gyro -frequency. The depth of

the impressed modulation was about 9%, and its time constant was

between 800 and 1200 |-Ls.

In 1957 a complete theory of gyro -interaction was developed (Snaith,

1957), which includes a rigorous discussion of the propagation of both

the wanted and disturbing waves. For a model of the night-time lower

ionosphere based on independent data, the resonance curve of interaction

computed using the complete theory agrees well with the observed

curves. The complete theory clarifies the physical processes occuring

in gyro -interaction and resolved in favour of Bailey the theoretical

arguments and experimental results obtained in England between 1947

and 1949 which caused J. A. Ratcliffe, D. F. Martyn, and others to

reject his theory. It also shows that the time constant of impressed

modulation is dependent on the frequency of the disturbing gyro -wave

and is a minimunn. when this wave is an exact gyro -wave. This behaviour

of the time constant was subsequently found on re -examination of the

experimental data.

The work in the period 1950-1957 demonstrated that wave interaction

in connplex situations involving oblique propagation can be predicted

with satisfactory accuracy.

1959 - 1963

The precise studies of ionospheric wave interaction experiments

using vertically propagated disturbing and wanted waves have the

considerable advantage over the gyro -interaction type of experiment

that the propagation of both waves, being at a fixed angle to the earth's

magnetic field, is much simpler to analyse. Also, the interaction

is free of effects from selective fading.

The equipment for the pulse experiments is located at three stations,

40
referred to as A, B, and C. Of these, A is the station from which

the earlier disturbing transmissions were made. The stations B and

C were constructed under Contract AF19(604) -6177 with the Geophysics

Research Directorate, Air Force Cambridge Research Laboratories,

Bedford, Mass.

Station A The gyro-transmitter at this station is a self -oscillator

having a frequency range of 1. 2 to 1. 9 Mc/s, a maximum pulse power

of 200 kw, and a maximum pulse duration of 1 ms. Its aerial is a circularly-polarized array of 4 horizontal half-wave dipoles arranged

in a square pattern. The disturbing beam from this installation is

estimated to heat electrons at night to a temperature of about 1000 °K,

i. e. , to about five times their undisturbed temperature.

The station has conaplete radio probing and receiving equipment

for wave interaction experiments of the pulse type.

Station B This station is a high power transmitting installation.

The transmitter is x-tal controlled for the frequencies 1. 43, 1. 515,

and 1. 60 Mc/s. The maximum power is 500 kw for CW operation and

1 MW for pulse operation. At the power of 500 kw, the pulses may

have any duration from 50 |Js to CW. The aerial is a circularlypolarized array of 40 horizontal half-wave dipoles suspended in a square pattern from twenty-five 90 ft. wooden poles. The array,

which covers an area of 70 acres, is stressed for a power of 4 MW.

The installation is estimated to heat electrons at night to energies

approaching 1 ev.

Station C The station contains extensive radio probing and

receiving equipment for the experiments with station B. It can,

however, also be operated synchronously with A. The station is

1. 5 km from B and 9 km from A.

The wanted frequencies used in the pulse wave interaction

experiments were 1. 78 and 2.12 Mc/s during the night, and 2.12 and

3.85 Mc/s during the day. The night-time wanted frequencies are

sufficiently close to the gyro -frequency of 1. 515 Mc/s for the

modulation depth impressed on the extraordinary mode to be significantly

greater (by factors of 20 and higher) than on the ordinary mode.

The experimental data obtained since 1959 using the pulse methods

are almost an order of magnitude more accurate than the gyrointeraction observations. Effects have been studied over a range of

a
power flux of the disturbing wave of four decades and decay processes

to six times the time constant. Accurate determinations have also

been nnade of the values of the wave interaction coefficient G and

electron collision frequency in the lower ionosphere. The most

important results are now summarized;

(i) Small-perturbation wave -interaction theory (i.e. , theory

applicable when the change in temperature of the electrons is small

compared to their undisturbed temperature) predicts that the depth of

impressed modulation is proportional to power of the disturbing wave.

It has been found that this linear relationship holds for changes up

to three times the undisturbed temperature. For larger changes of

electron temperature, the departure from proportionality to power is

not so marked as to cause appreciable error in an order of magnitude

estimate of the interaction.

(ii) For theoretical calculations of non-linear phenomena, it is

more accurate to use generalized magneto -ionic theory (Sen and

Wyller, I960) than the well-known theory due to Appleton and HartreeHowever, the errors due to the latter theory are not large enough to

justify revision of published calculations in which it has been used.

(iii) At a height of 8 5 km above sea level, the electron relaxation

time T, the collision frequency v in the Sen and Wyller magnetorn ionic theory, and the energy coefficient G (G is defined by T = ^

5"^ .3

1/G mmu ) have the values 640 [as, r- 2. 3 x 10 , and 6. 8 x 10 ,

, , respectively. For radio wave interaction interpreted using the

Appleton-Hartree fornaulae, the corresponding values of collision

5 -3

frequency and G are 7.8 x 10 and 2. x 10 , respectively.

(iv) At a height of 70 km above sea level, the values of T and v

6

^

are 40 ^ s and 4. x 10 , respectively. The Appleton-Hartree collision

7 frequency at this height is 1. 25 x 10 .

(v) The values of G ( or G ) have been found to depend on the m

temperature of the gas. For the variation of temperature with

height in the lower ionosphere measured using rockets, the change in

G ( i^ 10%) is small enough to be neglected in raost calculations of

wave interaction.

42

J
(vi) An electronic synthesizer for interpretation of D-region wave

interaction data has been constructed. Extreme accuracy of measure - ment is required before definite conclusions on the structure of the

D-region can be reached.

(vii) The modulation impressed on partial echoes from the Dregion has been found to consist of two parts, the first being the

normal interaction below the reflecting discontinuity and the second

to a change in the reflecting power of the discontinuity. The latter

is a new and hitherto unpredicted wave interaction phenomenon

with great promise for the study of the lowest levels of the D-region.

References

Bailey, V. A. (1937), Phil. Mag. 23' ^29.

Bailey, V. A. (1938), Phil. Mag. 26, 425.

Bailey, V. A. , R. A. Smith, K. Landecker, A. J. Higgs,

and F. H. Hibberd (1952), Nature 169, 911.

Fejer, J. A. (1955), J. Atmosph. Terr. Phys. 7, 322.

Sen, H. K. , and A. A. Wyller (I960), J. Geophys. Res.

6^, 3931.

R. A. Smith

Associate Professor

University of New England

7/1/64

it3

'An experimental study of gyro interaction in

the ionosphere, at oblique incidence"

F. H. Hibberd

Ionosphere Research Laboratory

The Pennsylvania State University

and

University of New England

Armidale, N.S.V/., Australia*

This paper describes further experiments on the interaction

between a gyro disturbing wave and an obliquely incident wanted

wave in the night time lower E region. The results fully confirm

those reported some years ago concerning the enhanced effect at

the gyro frequency. Average values of G v of about 800 sec were

obtained, with a height variation consistent with a scale height of

the order of 8 km. The effects of multipath interference and the

problem of the energy loss factor G are described in appendices.

*Permanent address

45
1. Introduction

This paper describes an experimental study of gyro interaction in

the ionosphere carried out in Australia by the author, R, A. Smith and V. A.

Bailey. Some results in the early part of this study were reported briefly

by Bailey et al.^(195Z)f. We present here an account of further experiments,

with improved precision, that were made following this eaTly work. The

experiments were designed primarily to test the gyro resonance effect predicted by Bailey, and were carried out at night using a pulsed gyro disturbing

wave and an obliquely incident wanted wave.

The resonance effect in gyro interaction depends in part on the well-known

resonance in the absorption of the extraordinary component of the disturbing

wave at the gyro frequency and in part on the path of the wanted wave in the

disturbed region. The phenomenon was analysed in detail by Bailey (1937, 1938).

Although the full mathematical description is somewhat complex, the basic

physical ideas can be presented quite simply as follows. Because of the very

high absorption near the gyro frequency, the energy in the extraordinary component of the disturbing wave is practically all absorbed after it has travelled

some 6 or 7 km upward into the night-time ionosphere. The region of the

ionosphere in which the electron energy is perturbed thus consists of a

relatively thin horizontal slab, above the disturbing transmitter, of fairly

large horizontal extent. The increase in electron energy within this slab is

much greater than that which would be produced in the same region by a wave

of the same power whose frequency is far removed from the gyro frequency.

The way in which the energy I absorbed her unit volunne in the slab varies with

frequency and with height in the slab is shown in Figure 1. At the lower levels

the curves have a maximum at the gyro frequency; at the higher levels there is

a subsidiary minimum here because the wave has been severely attenuated

before it reaches these levels. The curves in Figure 1 have been computed^

for purpoeee of illustration only, for a typical model of the lower night-time

he

i
ionosphere in which the electron density N and collision frequency v vary

with height as N = N exp(0.8) and V = 1.2 X 10"^ exp (-0.16 h),

where h is the height in kilometers measured from an arbitrary base height - -3

the bottom of the slab - at which N is taken as 1 electron cm o When a wantedwave travels horizontally through the slab at a given

height, the modulation that it acquires over an element of its path length will

vary with disturbing frequency in the same manner as the curve in Figure 1 that

corresponds to that height. Because the wanted wave takes a finite time

to pass through the disturbed region^the ^hase of the modulation acquired over

each successive element of path changes as the wave proceeds. But provided

that the time to traverse the whole path is small compared with the time

variation of the collision frequency , this phase change will be small and may

be neglected. Further, the path within the slab is not exactly horizontal in

practice but is slightly concave downwards and so the wave acquires its

modulation over a narrow range of heights with the slab. The result of this

is that the impressed modulation corresponds to average values of the

ionospheric properties over a narrow height range, rather than to values at

a sharply defined height. For the very obliquely propagated wanted wave used

in our experiments_,this height range is probably two or three kilometers.

From the foregoing it will be seen that a large amount of modulation

can be impressed on a wanted wave by a gyro disturbing wave when the wanted

wave travels nearly horizontally through the slab and so has a long path length

in the disturbed region. It is this combination of a gyro disturbing wave, with

its concentrated and strongly frequency-dependent effect, and an appropriate

trajectory for the wanted wave, that gives rise to the "resonance" in gyro

interaction. This situation should be contrasted with those in which (i) an

oblique wanted wave is reflected well above the slab and (ii) a vertically

incident wanted wave is reflected above the height where most of the energy

^7
has been absorbed from the disturbing wave. In neither of these situations

would one expect to observe a large effect at the gyro frequency or a marked

dependence on disturbing frequency.

2. Experimental Details

In the experiments described here the wanted transmitter was located

at Brisbane and radiated a continuous wave. The wanted wave was received

at Katoomba, west of Sydney and 720 km south of Brisbane. The disturbing

transmitter was located at Armidale on the mid-point between wanted

transmitter and receiver. It radiated rectangular pulses on a frequency that

was variable about the gyro frequency. Details of the transmitters are as

follows:

Wanted transmitter: Continuous wave, frequency 590 kc/s, power 10 kw.

Disturbing transmitter: Pulse length 1 milisecond, pulse

repetition frequency 40 per second, peak power 36 kw,

frequency variable in steps of 30 kc/s from 1390 to

1690 kc/s, half -wave dipole antenna. Local gyro frequency

at height of 90 km is 1530 kc/s, magnetic dip angle is 60 .

Observations were made between 0100 and 0220 hours. The band -width

of the wanted-wave receiver was sufficiently wide to avoid distorting the pulse

modulation acquired by the wanted wave. Automatic gain control was used in

the receiver, and a meter was incorporated in the A.G.C. circuit to indicate

the strength of the received signal.

The disturbing transmitter was switched on for one minute on each of

the thirteen frequencies irom 1390 to 1690 kc/s in succession, with a blank

interval of one minute between each disturbing radiation. During this blank

interval the wanted carrier was modulated for 30 seconds at its transmitter

with an 80 c/s sinusoidal signal to a depth of 5 per cent. This sinusoidal

modulation provided a calibration for the measurement of the depth of the
modulation impressed on the wanted carrier by interaction.

The rectified output of the receiver was applied to the vertical

deflection plates of an oscilloscope^ and the length of the trace photographed

on horizontally-moving film. Examples of the records obtained are shown

in Figures 2(a) and 2(b). Records of this type yielded the amplitude of the

impressed modulation as a function of the disturbing frequency.

The rectified output was also displayed on a second oscilloscope

with linear sweep and photographed at intervals of 3 or 6 seconds. The signal

strength was also recorded on this photograph. Figure 3 is a typical picture

of the display. The upper trace shows the increase and decay of the

incremental absorption of the wanted wave and hence of the electron collision

frequency. The shape of these pulses was used to obtain the time constant

for the change in collision frequency. The lower trace shows the disturbing

pulse which was received directly on a separate receiver after reflection from

the ionosphere, together with numerous multiple reflections. This trace plays

no essential part in the observations and was used merely to monitor the

disturbing transmitter.

The disturbing transmitter was correctly matched to the antenna at

each frequency in turn, and the antenna current measured with calibrated

meters. The various matching and coupling settings at the transmitter had been

deternnined by prior calibration. The power actually radiated was calculated

from the antenna current and the measured radiation resistance at each

frequency.

3. Experimental resonance curves

Because of the effects of nnultipath fading, meaningful measurements of

the amplitude of interaction can only be nnade when the received wanted wave

consists solely of the ray that has been reflected in the vicinity of the disturbed

^9
region. Fluctuation of the amplitude of the rectified pulse on the wanted wave and

occasional inversions of the pulse occurred sporadically for more than one

third of the total observing time. These fluctuations were always accompanied

by a decreased and fluctuating signal strength of the wanted wave. When the

wanted signal remained strong and reasonably constant (free of the deep fading

characteristic of multipath fading) the amplitude of the rectified pulse was

remarkably constant and records like those in Figare Z(a) were obtained. On the

other hand, when the signal strength fell or fluctuated, the records were like

those in Figure 2(b).

For reasons given in Appendix l.it was concluded that when the records

were like those in Figure 2(a), which were obtained when the wanted wave was

strong and steady, the only ray present was that which had been reflected once in

the E region. Records of this type alone were selected for amplitude measure

ment. The depths of impressed modulation obtained at each frequency were

normalized to a standard disturbing power of 36 kw. and plotted as a function

of disturbing frequency. (Some measurements of the depth of modulation were

also made on records of the type of Figure 3. These agreed well with the main

measurements)

During the 23 minutes required to traverse the range of disturbing

frequencies some multipath fading invariably occurred. In fact, on some nights

virtually no useful amplitude measurements were obtainable. It is usually not

possible to combine data obtained at widely separated times, because minor

changes in the gradient of electron density or- of reflection height produce

changes in the detailed shape of the resonance curves. Nevertheless, there

were sufficient occasions when complete, or almost complete, resonance

curves were obtained. Examples of such curves are shown in Figure 4.

It is s^en that these curves exhibit the predicted variation with

disturbing frequency and that a relatively large depth of nnodulation, of the

order of 5 per cent, is produced with a disturbing power of 36 kW. They

I

50 i
thus confirm in detail Bailey's prediction of the resonance effect. The same

conclusion was reached in the brief report by Bailey et al.j (195Z) of the earlier

measurements. The earlier measurements were made on the amplitude of

individual pulses, like that in Figure 3, and it was not always as easy to recognize

the occurrence of multipath fading as it was in the present experiments.

4. The time constant l/(Gv)

According to the conventional theory of wave interaction (Bailey and

Martyn^ 1934 J Bailey 1937, 1938J the time constant describing changes in

electron collision frequency v is given by l/(Gv), where G is a constant related

to the mean fractional loss of energy by an electron in a collision with an air

molecule. Thus the change in collision frequency produced at a point in the

ionosphere by a rectangular disturbing pulse of duration T varies with time as

1 - e"^^^ for O < t < T

and as e" for T < t < co The change in absorption of the wanted wave varies in the same manner. The

pulses of modulation impressed on the wanted wave should therefore consist of

an exponential rise and decay. This has been found to be the case, as illustrated

in Figure 3

A number of the clearest photographs of pulses in which the noise level

was low were very carefully measured, and the shapes were found to be indistinguishable from a pure exponential. Also, no significant difference in

exponent could be detected between pulses obtained in the absence or presence

of fading, including such severe fading that the rectified pulse was inverted.

It is shown in Appendix 1 that no difference is to be expectedjSnd for this

reason the measurement of G v was not restricted to those occasions when there

was no fading. Of course, it can occasionally happen that the relative amplitudes

51
and phases of the interfering components are such that the resultant wave is

over-modulated, but this produces a characteristic distortion of the pulse

shape which is very easily recognized. It occurred on only a very few occasions.

In order to measure the values of Gv from a large number of photographs,

an exponential pulse was generated from a rectangular pulse of 1 millisecond

duration and of adjustable amplitude, by means of a simple resistance -capacitance

arrangement. The generated pulse was displayed on an oscilloscope and was

matched in amplitude and shape to that in the photographic negative. Gv was

then obtained from the corresponding value of RC. The main limitation on

accuracy in the determination of Gv arises from the presence of noise on the

wanted signaL The uncertainty in individual measurements of Gv is of the order

of 10 percent.

Values of Gv were obtained from all records in which the noise level

was sufficiently low, irrespective of whether the occurrence of fading prevented

useful amplitude measurements from being obtained. The measured values of

Gv lay between 650 and 1000 sec" j and within these limits there was considerable

scatter at any one disturbing frequency. However, following the suggestion of

R. A. Smith, when the values of Gv were averaged separately for each disturbing

f requencyj a variation of Gv with disturbing frequency was found. This is shown

in Figure 5. Since the path of the wanted wave is independent of disturbing

frequency we conclude that this effect is associated with the fact that more energy

is absorbed lower in the slab at the gyro frequency than at disturbing frequencies

removed from the gyro frequency.

We can miake an estimate the order of magnitude of the scale height

from this. By replotting the data in Figure 1 to show the variation with height

of the energy absorbed per unit volume for various constant frequencies, it is

found that its maximum (or mean) value at the gyro frequency occurs at a height approximately 1 kilometer below that of the maximum (or mean) value

tor frequencies that are ± 150 kc/s from the gyro frequency. The scale height

52 I
H is

H = - V Ah/Av.

From Figure 5 we have that Av/v -^1/8 and, from the above. Ah "^ 1 km. This

gives H ~ 8 km,. within a factor of about 2. The effect seen in Figure 5 is thus

consistent with the known height dependence of the collision frequency at

altitudes near 90 km.

With the assumption that the time constant can be identified with 1/Gv , _3 we adopt the value G = 1. 70 x 10 suggested in Appendix 2. From Figure 5 we

see that Gv = 880 when the disturbing frequency is very close to the gyro

5 frequency. This yields v = 5.2 x 10 for the collision frequency. There is some

uncertainty about the height that this corresponds to but it must be where the

_3 electron d ensity has a value of 100 - 400 cm and is probably close to 90 km.

This value of collision frequency naay be compared with that obtained by

5 extrapolating Kane's (1959, 1962) data to 90 km, which is v a 5.0 x 10 using the

5 Appleton-Hartree magnetoionic theory or 2.4 x 10 using the generalized magnetoionic theory. The agreement suggests that the assumption concerning the

electron energy loss, discussed in Appendix 2, is at least approximately

correct.

53
Appendix 1

Effects of multipath interference fading

In oblique incidence transmission a wave may propagate from

transmitter to receiver by several different paths. For the experiments

described here it is necessary to examine the result of the simultaneous arrival

at the wanted receiver of a wave that has undergone a single reflection in the

vicinity of the disturbed part of the E region and other waves, such as those

reflected once or more from the F or E layers, that have not traversed the

disturbed part of the ionosphere. The ray that traverses the disturbed region

acquires modulation there by interaction; the other rays remain unmodulated.

Interference between the modulated and unmodulated continuous waves produces

a resultant modulation whose amplitude and phase depend on the relative annplitudes

and phases of the two components. The effects of sinnultaneous reception of

modulated and unmodulated rays can be seen by considering what happens when

their R.F. carriers arrive (i) in phase, and (ii) 180 degrees out of phase.

When the carriers are in phase they add, and the modulation depth

on the resultant is less than the true depth of modulation impressed on the

modulated component by interaction. However the phase of the modulation, or

for pulses ; the pulse shape^is unchanged by the presence of the unmodulated component.

V/hen the carriers are antiphased.they subtract. There are then three

possibilities:

(a) The amplitude of the unmodulated component is less than the

annplitude of the modulated connponent at the modulation troughs: The resultant

modulation depth is greater than that of the modulated connponent alone, -but

the phase or shape of the modulation is unchanged.

(b) The amplitude of the unmodulated component is greater than that of

^
the modulated component at the modulation peaks: The resultant modulation

depth may take any value. The phase of the modulation is shifted by 180

degrees. For pulse modulation this implies that the rectified signal will be

Inverted!, but the shape of the pulses will be otherwise unchanged.

(c) The relative amplitudes of the components are intermediate between

these in (a) and (b): The resultant nnodulation depth exceeds 100 percent, so that

the modulation is distorted.

It is thus seen that the true depth of the modulation impressed on the

wanted wave can only be obtained when the wanted wave consists solely of the

ray that has been reflected in the vicinity of the disturbed region. On the other

hand, except when the modulation is distorted (Case (c)), the presence of some

unmodulated component does not prevent meaningful measurements of the

modulation phase, or pulse shape, from being made

It is interesting to note that Ratcliffe and Shaw (1948) studied the

dependence of the amplitude and phase of sinusoidal modulation impressed on

the wanted wave as a function of modulation frequency of the disturbing wave.

They found that the phase measurements agreed much better with theory than

did the amplitude measurements and were unable to explain this. The observation

could well be accounted for if from time to time in their experiments some

significant amount of unmodulated wanted wave was received.

55
Appendix 2

The eJectron energy loss relation and the value

of G for air

The problem of the loss of energy by electrons in collisions with

air molecules has received considerable attention in recent years but it cannot

yet be regarded as satisfactorily resolved.

It is desirable, however, to interpret the values of the time constant

for changes in the electron energy, as measured from the changes in absorption

of the wanted wave. This will be done by retaining the original hypothesis that

the mean loss of energy by an electron in a collision is proportional to the

difference between the mean energy of the electrons and the mean energy of the

air molecules. When this assumption is made the time constant is identified

as the quantity l/(Gv), where G is the constant of proportionality in the above

hypothesis

Information about the hypothesis and the value of G, if it is in fact a

constant, can be obtained from laboratory experiments on the diffusion and

drift of low energy electrons in gases. Unfortunately, the experimental data

for air only extend down to mean electron energies that are about six times

greater than thermal and extrapolations or other assumptions must be made in

order to obtain values for the lower energies relevant to ionospheric studies.

We introduce the following symbols used in diffusion and drift studies:

Q = mean energy of agitation of a gas molecule

Q = mean energy of agitation of an electron

k = Q/Qo AQ = mean energy lost by an electron in a collision with a gas molecule

X. = AQ/Q = mean fraction of its energy lost by an electron in a collision

Z/p = ratio of applied electric field to gas pressure

56
The quantity k represents the mean electron energy expressed in terms

of the energy of a molecule as unit. The product \ k represents the energy lost

by an electron per collision, in the same units, and is directly related to the

drift velocity of electrons in the gas . For a given gasjk and \ k are each

functions of Z/p alone and may be obtained from separate laboratory experiments

By associating corresponding values of k and X k measured at the same Z/p

one obtains X. k as a function of k. The most reliable values of k for air are

those obtained by Crompton, Huxley and Sutton (1953) and, of drift velocity and

\ k, those obtained by Nielson and Bradbury (1937). The values of Xk as a

function of k, for the lowest energies, are plotted in Figure 6. The lowest

energy represented here is k = 5.9, whereas the region of interest for

ionospheric effects is near k = 1. The original hypothesis of Bailey and Martyn (1934), based on measurements in air by Townsend and Tizard (1913) and on the theoretical

behaviour for elastic collisions, is that the mean loss of energy by an electron

in a collision is proportional to the difference between the mean energy of the

electrons and that of the gas molecules . The hypothesis may be written as AQ = G(Q - Q^)

when G is the constant of proportionality. This is equivalent to

X k = G(k - 1)

If the hypothesis is correct the graph of Xk against k should be a straight line

which intersects the k axis at k = 1. In Figure 6 the expe rimeii al points from

k = 21. 7 to k = 10. 5 lie very closely on a straight line whose extension meets the

_3 k axis at k = 1. The slope of this line yields for G the value of 1. 70 x 10

(Points for k greater than about 22 lie above the line^but this effect is presumably

associated with electronic excitation and is not relevant to our discussion. )

57
The fact that the five points below k = 10. 5 fall increasingly below the line

have led to serious doubt about the validity of the G hypothesis for air. It

should be noted that the greatest deviation from the line, shown by the lowest

points, would correspond either to a deviation of 20 per cent in the value of the

drift velocity or a deviation of 45 per cent in the value of k. Whether the

results of Figure 6 either establish or disprove the G hypothesis is far from

obvious, as indeed also is the more fundamental question of whether there

is any thermodynamic or other reason to expect the hypothesis to hold for gases

other than the rare gases.

Data for nitrogen are available for considerably lower energies than

those for air and arguments have been'given that suggest that at very low

energies the loss of energy by electrons in collisions with oxygen molecules is

much less than those with nitrogen, so that the energy loss in air at very low

energies may be estimated from the experimental results for nitrogen. On the

other hand the results of Brose (1925) and of Healey and Kirkpatrick (1939) for

oxygen at k '^ 6 indicate that at this energy the energy loss is much greater

in oxygen than nitrogen. There is still uncertainty about these questions. If,

for sinnplicity, one adopts the G hypothesis for air as a tentative approximation,

the best value for G would appear to be 1. 70 x 10

Finally, it will be I'ecalled that it was at one time proposed that the

2 energy loss in air was proportional to ( Q - ^ )

. This relation was subsequently

withdrawn because it was considered to conflict with deductions from the magneto

ionic theory. This discarded hypothesis is only mentioned here because it has

been revived in a recent report. A simple argument shows that the relation

cannot be true in the vicinity of k = 1. Because the velocity distribution of

each set of particles, electrons and molecules, is Maxwellian or very nearly

i

56
so near k = 1, the suggested relation is equivalent to the statement that the

2

electron energy loss (or transfer of thermal energy) is proportional to (T - T ) where T is the electron temperature and T is the gas temperature. This

implies that the direction of heat flow does not reverse when the temperature

difference changes sign, which contradicts the second law of thermodynamics.

The electron energy loss must clearly be an odd function of Q - Q in the

vicinity of k = 1.

The uncertainty about the dependence of the electron energy loss on

the electron energy constitutes one of the most serious problems in the theory

of wave interaction. It is directly related to the other outstanding problem of

the manner in which electron collision frequency depends on the electron energy.

References

Bailey V. A. (1937) Phil. Mag. 23^, 774 and 929.

(1938) Phil. Mag. 26, 425.

Bailey V. A. and Martyn D.F. (1934) Phil. Mag. 18_, 369.

Bailey V. A., Smith, R. A., Landecker, K. , Higgs , A.J. and Hibberd, F.H.

(1952) Nature 169, 911.

Brose, H. L. (1925) Phil. Mag. 50, 536.

Crompton, R. W. , Huxley, L.G.H. and Sutton D. J . (1953) Proc Roy.Soc.

A, 218, 507.

Healey, R. H. and Kirkpatrick, C. B. (1939), quoted on p. 94 of "The behaviour

of slow electrons in gases" by R. H. Healey and J . W. Reed,

1941, Sydney, A.W.A. Ltd.

Kane, J. A. (1959) J. Geophys . Res. 64, 133.

<196l)J. Atmosph. Terr. Phys._23, 338.

Nielson, R. A. and Bradbury, N.E. (1937) Phys . Rev. 51_, 69.

Ratcliffe, J. A. and Shaw, I. J. (1948) Proc. Roy. Soc. Aj^2' ^11.

Acknowledgement s This work was supported by the Radio Research Board of Australia and by

research grants from the Universities of Sydney and New England. The wanted wave

transmissions were provided by the Australian Postmaster General's Department. Preparation of this paper has been assisted by NASA under Grant NsG 114-61.

59
C

r-l

0)

u

X, 3

(-1

O T3 O O >

4)o •Sm

0) ^1 B >

\i >-, "^ *j

>H

C Ji QO ° <^ ?^ 'fll 9) 3 3 4)

(« 0^

Ji

w ^t ^ ^ m ^ _ O

*- 0,

O .^

>-

O n; •»-> 2 o

*^ -

"• ^2 •?: a, u

OB -^ ^ «J Ji <u

<"'

<« OB 4j 4J (B O > X o o 5 "Id o

t)

0)3

•f-<

60
c -v O 4) '^ «^

r^ a;

(I)

X o > ^ J=

(D "

•o rr

»i nJ rt ^ o o _^ n

"^ ^ -D

!5 ° o

S^ " "J ^ o V a.S ^ E ".2 c 2 ^ o 'Z! S "^ .

=> ;^ ::3 c O -M 5 B ^ o £ ^ 3 T3 C Of ;S S ° c « 2 >< - J o c « », u «

fCO (U -c

u

fe c

•D

n! 3EcC

XIo

n)u

kl3

&0ouv

kl

(ds•rH W

41M3

60 > ^

41

O ^ m C D 00 O (J 4) ^ .

»-

4)

_

a

'^

<U 41

-ir

-^ 4) 3 —^ fli O 3 cfl

a. ^ Bc

«, -S c

x; " J3

*^ T3 ki

n) to

So' 2 2! u V ^

61
10 r

M7o

/ o

lOr

M%

10 r

M%

1.4 1.5 1.6 1.7 DISTURBING FREQUENCY Mc/8

I

1.4 1.5 1.6 1.7 DISTURBING FREQUENCY Mc/s

Figure 4 Experimental curves obtained on various occasions, showing the

depth of nnodulation impressed on the wanted wave as a function of

disturbing frequency.

62
10

liJ

P, ^

UJ ti m

a:

10nipO>

63
MEAN

ELECTRON

ENERGY

LOSS

PER COLL.

XkxIO^

MEAN ELECTRON ENERGY k

Figure 6 Experimental values for air of the mean energy lost by an electron

per collision, X k , .as a function of the mean electron energy, k ; after Cronnpton et al (1953). The energies in terms of the mean

molecular energy as unit. The straight line corresponds to the hypothesis that the electron energy loss is proportional to the difference between the electron energy and the molecular energy.

4

6k
NONLINEAR INTERACTION OF ELECTROMAGNETIC WAVES

WITH MAGNETOACTIVE PLASMAS II*

Mahendra Singh Sodha and Carl James Palumbo

Republic Aviation Corporation

Farmingdale, New York

In this paper an expression has been derived for the

nonlinear conductivity tensor of a Lorentzian magnetoactive

plasma which is correct to quadratic terms of the amplitude

of the elective vector. This expression has been substituted

into the wave equation and the resulting nonlinear differential

equation has been solved to the second order of approxinnation. Using this technique the propagation of electromag|ietic

waves in an infinite nonlinear magnetoplasma and the-reflection and refraction at the interface of a magnetoactive plasma

and a linear isotropic medium have been investigated.

Boltzmann's transfer equation for electrons in a uniforna plasma

may be written as

df ,. df / df / il _ ( ^

¥t ^ °^x "3^ ^ °^y 3u ^ ^z bv " Ut

X y

z

* (1)

where the symbols have usual meanings.

The acceleration a

' of electrons in the presence of an electric

field E^ = E exp (i uu t) and magnetic field jB is given by

- a' = a ' exp (i mp,t) + u x uu (2)

where

(JU = -^

,

The full version of the paper has been published in Canad. J. Phys. , 42, 349 (1964).

65
a = , -'^ m

e is the electronic charge,

m is the electronic mass,

and c^ is the velocity of light in vacuum.

For a Lorentzian plasma it may be shown that [ Chapraan and

Cowling, 1939]

\-t\ = -^<^- V + —T — <^o" ^^ + —2 h" —I- ('^

\ 'c Mu Su My \

The symbols have usual meanings.

Proceeding in a manner similar to that given by Sodha [1960],

J^

pT^ =(CT e + a & +a £) exp(i uu„t), XX X xy y xz z' -^ ^ {(J&+0&+0&) exp(i uu.t), (4)

J

QQ yx X yy y yz z' ^ '

J

= (^_^^ + ^.,A. + ^„„^J exp(i m t). E Q zx X zy y zz z

where J is the current density,

E is an arbitrary normalizing field,

00 ^ 5 »

En

^ = T^—

-^ E

00

The components of the nonlinear conductivity tensor o may be

expressed as

66

J
%6= %6„ + ^i^s/^-^'+^s/fS'""^ <"

2 2

where a = 6m k T

p, 6 = X, y, z,

$(e, 0)) = (£ . UJ) (£ . uu),

and O ^ , O ^ , and cr ^ can be expressed in terms of known integrals

p6q p6^ p6^

involving collision frequency.

Substituting E = E^ exp (i uu t) in the wave equation, and express

ing the wave equation in dimensionless form, one obtains, for propagation along the Z axis,

d^e r , . —^ = -^^ < -e + ^^

(a e + o e +a e)V, (6A)

d§ ^0^0 I

"" GUJq XX X xy y xz z >

d^e^ - -^^ ^-e +^^(ae+ae+ae)V, (6B)

d§^ ^0^0 1 ^

^"^0 ^^ ^ ^^ ^ ^^ ^

d^e

2

and —^ = -3 £ (6C)

where f = ^^o^'o^ '^O^

^0

67
and 3^ = -21L ji _ IHI ^ . iHi [ ''zx^y ^ V^yh (6d)

The electric vector may be expanded as

& = e' + a^&\ (7)

2 Combining (6A), (6B), {6D), and (7), putting 3 =0 (following the usual

z procedure [Ginzburg, I96I]) and equating like powers of a , we see that

the propagation of the two modes £ + a £ and £ + a £ is described by ^ ^ ^ xlyxZy '

2

-^ {£"+a,£'') + 3;(£''+a,£'') (9A)

^^2 X 1 y 1 X 1 y

= (a £' + a £') £• £ + (b £' + b'£') $(£, uu),

^ 1 X ly'-^-- Wx ly'^

and ^

(^; + -2^;) + 33^£;.a3£;) (9B)

= (a £' + a '£')£ £ + (b^£ ' + b'£') $(£,0)), 2x Z y' '^ ^ ^2x ly ^'^

where a^, a^, 3 , 3 a , a , a' a' b , b , b' and b ' can be expressed in terms of a , a , and a

68
2

The expansion is valid when i|/ << 1. Significant nonlinear

effects will occur when i|/ is not much less than unity. Hence, the

theory is both useful and valid ordy in the domain when i|/ is not much

2

less than unity, but i|/ is much less than unity. Even in the case when

2

t is not much less than unity, the theory should give qualitatively

correct results.

For a given value of the amplitude of the electric vector

E , f is minimum when the magnetic field is along the direction of

propagation, because in this case ^ • uu = 0. Hence, for a given

combination of the amplitude vector E and the magnetic field B (or uu),

the nonlinear effects are least pronounced when the direction of propagation coincides with the direction of the magnetic field.

When the direction of the electric vector coincides with that of

the magnetic field, (E„ * uu) and hence t has a maximum value for a

given combination of the amplitude of the electric vector and the magnetic field. Thus, in this case the nonlinear effects are inost pronounced.

It may be added that in this case the propagation is in a plane polarized

mode in contrast to two elliptically polarized mode propagation in

general

It may also be seen from the present analysis that the two modes

of propagation of electromagnetic waves in a magnetoplasma do not

propagate independently of each other; the complex amplitude (and hence

69
The solutions of (8A), (8B), (9A), and (9B) may be obtained by

using the radiation condition and the boundary conditions

X xO y yO

£ = e = X y

at § = 0,

at § = 0,

and hence

The problem of reflection and refraction of an electromagnetic

wave at normal incidence on the interface of a linear isotropic medium

and nonlinear anisotropic medium has also been solved, and explicit

expressions for the reflected and refracted components have been

obtained.

The second order nonlinear theory presented in this paper is

valid when the expansion

(1 + t)"^ « 1 - t

is valid, where ^ is given by

e 2/2 M / 6m k T

r 2 , .2-, r 2 2^

Lv + (o) + (JO ) ] Lv + (uu - (ju ) J 222 2 2 2 ~ (v+O)- 3«j ) ^0 • ^0 <^ + % + "^ ) + (^^ • ^) (E^ • ^

(V + 0)^)

^

TO
the intensity and phase) of a given mode does depend on the complex

amplitude of the other mode. This nonlinear interaction of the two

modes will give rise to cross -modulation, if both of the modes are

amplitude modulated in a different manner.

References

Chapman, S. and T. G. Cowling (1939), Mathematical Theory of Nonuniform Gases, p. 348 (Cambridge University Press, New York).

Ginzburg, V. L. (I96I), Propagation of Electromagnetic Waves in

Plasma (Gordon and Breach, New York).

Sodha, M, S, (I960), Electrical and thermal currents in a slightly

ionized gas, Phys. Rev. , 1 19 , 882.

71
Collision Effects in Hydromagneto- ionic Theory

Hari K. Sen

Air Force Cambridge Research Laboratories, Bedford, Mass

and

Arne A. Wyller

Williamson Development Co., Inc., W. Concord, Mass.

A hydromagneto-ionic theory has_been_ developed within the

framework of the Burgers formalism _/ 1958_/ which is a microscopic

theory based on moments of the Boltzmann transport equation. The

effects on electromagnetic wave propagation of electron-electron

and electron proton collisions have been considered to the order

of Chapman and Cowling's second approximation. The present theory

is an extension of the magneto-ionic theory derived earlier by one

of the authors /jWyller, 196X7 for a fully ionized hydrogen plasma,

in that it includes the effects of ion motions. The theory is

therefore applicable to the low frequency modes of wave propagation, such as the whistler, Alfven and the retarded magnetoacoustic modes. Expressions have been derived for the refractive

index, absorptivity, the wave polarization, and the zeros and

infinities of the refractive index. Numerical applications have

been given for the four characteristic modes of low frequency wave

propagation, viz., the whistler mode, lower hybrid frequency, ion

gyroresonance, and the hydromagnetic mode. Applications of the

theory to the solar corona, and future extensions to the terrestrial ionosphere have been indicated.

72
Collision Effects in Hydromagneto-ionic Theory

I. Introduction

The present paper considers the effect of the ion motions

on electromagnetic wave propagation in a uniform plasma with

a superposed homogeneous magnetic field. The conventional

magneto-ionic theory considers the motions of the electrons

only, which is a valid approximation for the high wave frequencies used in ionospheric propagation or for a weakly ionized

gas. When the wave frequency goes below the electron gyrofrequency, the extraordinary electromagnetic wave becomes

the oblique Alfven (whistler) mode, and for a further decrease

below the ion gyrofrequency the ordinary electromagnetic wave

becomes the retarded magneto-acoustic mode. Delcroix and

Denisse have given an excellent illustration of this in their

book [1961]. We note that if we take account of the pressure

tensor, two additional modes will appear, viz. , the electron

and the ion plasma waves. We have not considered these

modes in our paper.

73
Our work differs from most treatments [Schliiterj, 1950,

1951; Dimgey 1951; Astrbmjl951; HmeSjl953; Piddington^l956;

Gershman, Ginzburg^and Denisov 1957; Lehnertjl959; Fejer

1960; Delcroix and Denissejl961; AlliSjl963] in that it takes

account of coUisional damping by a microscopic kinetic theory

with a proper averaging over the velocity distribution function,

of like particle interactions (proton-proton and electron-electron collisions), and of the heat-flow equations.

We have seen that a microscopic treatment through the

Boltzmann equation led to significant departures in the propagation factors for electromagnetic waves in the terrestrial

ionosphere [Sen and Wyller_, 1950; Leinbachjl9 62]. Later on,

one of us [ Wyllerjl961 ] applied the Burgers formalism [1958]

to find the electrical conductivity for a completely ionized

hydrogen plasma. The Burgers formalism proceeds via

moments of the Boltzmann transport equation and the associated

collision integral. It includes the effects of like particle interactions and of the electron and ion heat flows, and is equivalent

to the Chapman and Cowling second approximation [1958], The

present work is an extension of Wyller's [1961] work to include

the effects of ion motions.

7^
In a recent work Kantor [1963] has considered the effects

of ion motions on magneto-ionic theory without collisions and

considered the properties of different directions of wave propagation with respect to the magnetic field. Our work checks

with Kantor 's formulae for the case of no collisions.

II. Basic Equations

It is well known that the presence of magnetic field in a

plasma introduces an anisotropy in the propagation of electronaagnetic waves through the medium. This anisotropy is

characterized by the tensorial representation of the associated

physical quantities governing the modes of propagation; namely,

the electric conductivity and dielectric constant of the mediiun.

After introducing the dielectric tensor (which is easily derived

from the conductivity tensor) into the Maxwell equations^one

can derive expressions for the refraction, absorption^and polarization of the propagating waves. With this end in view we shall

set up in this section the fundamental equations that will enable

us to compute the a. c. conductivity tensor of the magneto-ionic

medium which may transmit electromagnetic waves associated

with the oscillations of the ions as well as electrons. In doing

75
that we shall adopt, as we have already said. Burgers' formalism for d. c. conductivities in the a. c. case.

Choosing a rectangular coordinate system (o, x, y, z)

such that the z-axis lies along the uniform applied magnetic

field H , the electric conductivity tensor can be put into a

very simple form. The electric field due to the propagating

wave is represented by an a. c. field in that medium^

E cos out = (E T + E "j + E k) cos a)t»where i, j and k ji. y z are unit vectors along the x, y and z axes and uo is the excitation frequency of the wave which is assumed to remain constant during the propagation of the wave inside the medium.

For a simple model we shall assume the plasma medium

to be homogeneous with no pressure (and density) gradients.

We shall consider a fully ionized hydrogen plasma containing

electrons and protons in equal number densities, n^ = n

throughout the medLum. The subscripts 1 and 2 will be used

consistently to denote protons and electrons respectively in

the following discussion. Furthermore, under these conditions it will be appropriate to assume that the mean velocity

76
of mass flow of the plasma as a whole is zero. In terms of

Burgers' notation this would mean that the components of

diffusion velocities of the two types of particles reduce to

their respective mean velocities, i. e. ,

W<K -'tih . Wz.h - tz*i ) k=^t,i,3 - (1)

The assumptions m.ade above bring about a considerable

simplification in the Burgers formalism. There is a simple

relation between Burgers' "friction coefficient" Kiz - Kj_t -K

and the mean electron-proton collision frequency V , namely,

(2)

Here "Yp is the Debye length and Qi© is the electron plasma

frequency.

In order to assess the full implications of the momentum

and heat transfer equations of the electrons and protons, we

state here some of the physical parameters of Burgers'

77
theory that are relevant to the conditions of our problem.

For a binary gas with known relative velocities of the colliding particles "g" the collision cross section is given by :

where b and X are Chapman and Cowling's [1958] collision parameters. In particular, C - 1 for momentum transfer with which we are primarily concerned here. The average

collision cross section for all possible relative velocities is

o

Here the subscripts " s " and " t

" refer to the species of the

colliding particles and JU. is the reduced mass. The superscript "C" refers to the angular dependenccjwhile " J

" refers

to the velocity dependence of the collision cross section. In

terms of these average collision cross sections, the quantity

defined by

(/2)

Zst-1-|-^^ (5)

78
reducesto the value 3/5 for Coulomb interactions.

In order to be able to estimate the effects of collisions

between like and unlike particles separately, Burgers [Part II

pp. 65 and 93] introduces the following collision factors which

turn out to be proportional to the average collision frequencies,'

)C = ^ mimx. /^i2. (electron-proton collisions) j

>^, — 2 / ^fyK4-nn2.\ ^ L^n (proton-proton collisions), (6) (•

Xo ^/ Illldi?!!^^ / '7^?*' (electron- electron collisions)*

It will be seen later that the auxiliary quantity ^ which

incorporates the effects of electron-proton and electron-electron collisions enters into the electrical conductivity tensor.

It is defined by

%= X-fXx + yY2.7 '^'"^^ and (7)

12. is a function of

^

only.

79
Though the proton-proton collisions do enter into the

equations of the heat flow through the factor 3Cf , their net

effects in the electrical conductivity tensor vanish [Spitzer^

1956; Marshall^l957]. On the other hand, in thermal conductivity the dissipative effects due to ion-ion encounters is

not negligible [Rosenbluth and Kaufman jl 9 5 8; Wyller^l963].

Burgers' momentum and heat flow equations in their timedependent forms will be our fundamental set of equations for

the electrons and protons. We will rewrite Burgers' equations [Part 1, p. 20, eq. (37) and p. 42, eq. (83)].

Momentum equations for electrons:

Momentum equations for protons:

^ U) K/jx-Cx- SI Wi> = -2ri7(W»y-IVay)-E^VKx-4.?i^>;^ , (9)

80
Heat flow equations for electrons;

(10)

»

7 2. IQ

2. i^

Heat flow equations for protons:

In the above equations the quantity t/CO appears because

of the time dependence of the particle velocity and heat flow

velocity of the form \^e.^^ and ^(j>^^^\

respectively induced by the alternating electric field S = Co(q ^^)"

Also the accelerations due to the electric field

are denoted by

SO that

r --"^n 1 ^-wu" ' (13)

81
The gyrofrequencies are defined by

The above set of 12 equations in 12 unknowns shall now

be solved to obtain the current flow components ( Ia/^K— WzA ).

Then the conductivity tensor will be found from the usual formula:

(14)

III. Solution for Electrical Conductivity

Along the Magnetic Field

The momentum and heat flow equations for both electrons

and protons along the z axis are:

^ ^^ (16b)

According to Burgers we neglect the terms 9lfvl\%^

Binrtr* , and ^^'y/Tit for ryr««Ti«-

This can be verified by solving for Tf/^ and Talk from the

basic equations of the last section.

From (16a) we have

n^.^^^^vL^'*-h^4 . (17)

82
Upon inserting (17) into equations (15a, b) we find that

Defining the current flow velocity as 1^^= i/sIi—l^'Z. ^^

have

Thus we get from, equations (18a, b),

>LUjWa-i-(l+^)R:« =-0-f2^)V We4Cl-h^)f^^^Wz . (20)

Now solving for Ui» with separation of real and imaginary parts

we obtain

(21)

^cw)4-l^b^)-^-(j^f-^*>4^fc)>-o«r«-

At this stage we introduce simplifying notations to condense

the final expression for the conductivity tensor component along

83
the z- direction and to elucidate therein the effects of electronelectron and electron-proton collisions',we set

{;i+0-fy)|a] (22)

The component of the current vector in the z-direction is

then given by

and the conductivity tensor component in the z-direction is

finally obtained

When we compare this tensor component with the corresponding Lorentz tensor component ^/^-i-iCO) , we find that

the ion- motions affect the conductivity along the z-direction

through the factor (l+Jf) while the electron-proton and electronelectron collisions through S3 and V3 . In particular, the

expression (24) checks with Spitzer's d. c. conductivity [1962],

when"^ and CJ are set equal to zero.

Ok
IV. Solutions for Conductivities PerpendiciiLar

to the Magnetic Field

The magnetic field couples the x and"/ components of the

momentum and heat flow equations, making the solution for

the flow components WCx and Ujly somewhat awkward to

obtain. We may avoid this difficiilty by introducing a coordinate

system defined by unit bi- vectors [Menzeljl961].

£-k^i^^0 .j^-k(^--P^ ^-^.

(25)

Now we may express the electric field as, omitting the time

dependence for simplicity.

g^ €,1+ ^yf+§e^^L(i^-l6j)^yi^+i€y)p^ij^^ (26)

AUis used a similar coordinate system in his formulation

of the conductivity tensor for plasmas [AlliSjl956], eg corresponds to the left rotating field component and ^y. to the

right rotating component.

We also write the flow components in this coordinate system j

85
We begin by solving for Wr from the rewritten momentum

and heat flow equations'.

tU3Wzr-R.r-lSzWxT = vCWiT-W2j) +H V Tar (28a)

tajTtr -ISxYir - — Jh vCWir-U/nr)— f VfLr

(28b)

(29a)

LCoTTr-l-tS.rrr =|22rvCW.r-M-)^.l^«*ir +m^iJ-rxr-

^^^^^

From equation (28a) we have

TrCUi»r-Wxr) = lcoUl^r^fl.T''iS3.i^^r-sVYir ^^^^

which yields Txr when put into eq. (28a)

Tit - 5 ^r i ( S'L,-Co)VJxr -h Jlr J (31)

Putting expression (30) for V (Wir— U/i-t} into

equation (28b) we obtain

•^^ ^^^^'^-

(32,

since |7r =. —^ [ir . The electric field and heat flow terms

have dropped out. At this point we may return to equation (28a)

with these expressions for W/r and Tzr and solve uniquely for

86
The total flow associated with the right rotating electric

field is then the sum of the electron and proton flows

Wy = Uir-UIit =nrCs?r^>? ^VlUliT . (34)

For subsequent algebraic simplification we introduce the

symbols

which gives

(35)

Now, expression (35) is equivalent to

Ur=z ^ f±^i& -_ (36)

C I V + iCSi-uj)J C^ iT-t C Si.- cj) J

On separating the real and imaginary parts in the denominator

we have

Ulrs ^ ^ ('^^}hT (37)

Now we introduce a new symbol 6l and the expression for

Wr becomes

aCvO-^.)+L(cc>-s,)0+^) ' »•§ T/^U)^(38)

87
We can repeat the foregoing analysis and obtain the flow

for the case of the left rotating electric field

where^as before, we have simplified the expression with the

symbols

From the above, we obtain the total current

T= -nae^.CWcX'-fUJr/f+W.'f )^ (40)

Rather than directly solve for Wx and W^ , we derive the

electrical conductivity tensor in a form analogous to that of the

Lorentz tensor form [Sen and Wyller I960]. We first introduce

the symbols

-/et "llr -||^ (41)

which let us rewrite expression (40) as

T = ^^e^^j. tt fuiV^ L]^)+-^r.r.rCwpt ^»fP«t7(42)

and

88
or, in matrix form

(43)

Ix

8tr

V

o

J

(44)

Then the complete electrical conductivity tensor in the

hydromagneto-ionic theory becomes

(45)

<3 iy*s J

<^-r: ^f-K^^^-lTt) (^t+Tr)

where

(l+,/^£j[/V, + L(u)-sO]

'^

(l + /3 £2r) [(3 \^j. + C (cO+Si)j

(46)

T* =.

(fi-f^)

[l+Cl+^)^][(Hr}V3-htu>] '

89
Thus we have been able to express the electrical conductivity tensor containing ion-motion, electron-electron and

electron-proton collisions into a form which is similar to that

of the Lorentz tensor [Sen and Wyller^l9 60 a].

When y goes to zero, expression (45) reduces to a form

of the electrical conductivity tensor [ Wyller^l961 ] valid for

high frequencies (conventional magneto-ionic theory).

The dielectric tensor is derived from the relation

(47)

for which we find

i 2. em- La,v--^CCi\+e„)3-f'tCl>ii- -t(f,i4c^i«)3 (48)

where

f

Li+Ci^«|3][(Hjf)^4-^co^j "^ ''" c35T04aj^^H^)^J3?5

•^^1-.

(^9)

C„ ^ too ^y"?"^^^^ ^11 = coo f3 );fc

90
Retaining the Sen and Wyller formalism [1960] we find

that the formulae for the refractive index j^a. polarization

R are unaltered in form except that the constants A, B, C, etc.

are now functions of ^I , €n > ^^^ ^ Iff defined in the

equations (48) and (49). This leads us to a new formulation of

the hydromagneto-ionic theory which permits the study of low

frequency modes such as the Alfven mode and the retarded

magneto-acoustical mode with electron-electron and electronproton collisions included. For convenience we recapitulate

[Sen and Wyller, 1960 b];

(5D,a.)

Here y\ is the real refractive index, CX is the absorptivity

and <^ is the angle between the direction of the magnetic field

and the direction of propagation.

V. Zeroes and Infinities of the Refractive Index

In the formula (50) the complex refractive index Ti = ^r>- lc><

j

satisfies a biquadratic equation

cT :=(r\-iCK^ :=. A -<- B Svn> i: C B s;^<»- C"co?-») .53)

91
Hence the condition that n may have a zero root is that

provided that J>+e"Sm <^ r^s O

After some algebraic simplifications^ the above two expressions reduce respectively to

€i [r^i+em)'^ Gn ] (€i + €m ^^4>) = o (52)

(53)

so by virtue of the restriction (53) all the zero roots of fTwill

be given by the following three equations,

•(a) 61 =- O (b) €r+€a-t6a =

(c) Gr + €Qi -t-i-^n =© •

These equations are independent of the propagation angle "^".

Assigning the appropriate values to €1 from (48) and (49) we

find that (a) will hold when Q.n — j and tn =. O

simultaneously. The latter condition means that either v* — O

i. e. there is no collision or else that S3-=.\ , which gives

an imaginary value to the collision paranaeter 'Z — v^ •

The first condition with V rr q yields a cutoff for T) at

X = >

. <54)

""

/-far

We use the symbols X for OJo/cj^ and "Y for Sx/ti)

92
However this root is attained only in the absence of collision;

while we may say that collision removes the cutoff. Similar

treatment to equations (b) and (c) yields cutoff values for n

without collisions at

and '"^^

(55)

respectively. "*" ^

Analagous to the analysis of (a) these values are not attained

actually when collision is present.

These cutoff values will be illustrated (with proper qualifications) by the curves for n against x on pages ( er\d of (Rej^'c^'^^)-

By inspection of the eq. (51) it is apparent that (n)* will

have an infinite root when

"D-t- E Si^<p - o

This is equivalent to the condition

€i +€irS»^4*= O (56)

For the case of collision the condition (5 6) turns out to be fairly

complicated, but if we neglect collisions it yields a simple

relation analogous to that of Hines [1957].

93
It may be noted that the condition (57) for infinite root of n is

dependent on cj> . So if ^^TL^n will have an infinite root at

X = Ci-V^)(/-yV) . (58)

This value is somewhat different from that obtained by Budden

[19 61] even if an allowance for his approximation is introduced

in our computation. For low frequency end of the whistler

spectrum, we consider the case Sx»CO>>Si . In this limit the

condition (5 6) simplifies down to

Now if the coefficients of Cos ^ and Sv^ <^ in the equation (59)

are both negative then the condition for an infinity will not hold,

meaning thereby that the wave can propagate in all directions.

Analysis shows that the first coefficient will be negative if

Co<CJo ; also the second coefficient will be negative only

when o) < ^oj g'-i-^^^i. . These bounds on cd will be helpful

for the detection of ions with whistlers (^U)«.CJo ) which are

propagated to a large ,extent in the upper atmospheric regions

where our present model of a fully ionized plasma will have a

closer approximation [Hines^l957].

9^
VI. Numerical Applications

The real refractive index, TX , and the absorption factor,

ex , were evaluated from the relations

and thus rigorously

We note that our signs in equation [S^/ fo^ ^he complex

refractive index are opposite to those of the magneto-ionic theory.

We have computed the propagation constants VI and C x/cj as a

function of X for the following representative values of ^ :

(i) ^=|00 (whistler mode);

(ii) ^^yjl/Y (low^r hybrid frequency);

(iii)

I "= -^ (ion gyro-resonance);

and (iv) ^ =- \0 (hydromagnetic mode).

We will now briefly discuss the above cases.

Case (i) "^ =. loO (whistler mode)

Longitudinal propagation ( <^ = o)

(^Figures 1 and 2J

The introduction of ion motion has little effect on the birefringence,

which is only slightly decreased. The important parameter for

95
collisional effects seems to be the ratio Z/Y • Even for small

values of this ratio (0. 1 in Fig. 2), the cutoff in the ordinary

ray is removed. The absorption of the extraordinary ray is

increased with increasing 'Z, . The reverse is the case with

the ordinary ray.

Transverse propagation T

4* ~ ^i

)

. Figure 3. O3 is

not drawn in the figure. It follows very closely the vertical axis,

starting from the value n =. I oJl X=0, with a cutoff at X — 0. 995.

The effect of ion motion is negligible for the ordinary ray. It is

quite marked for the extraordinary. The birefringence of the

medium is appreciably decreased. The collisions remove the

cutoff for both the ordinary and extraordinary rays. The inclusion of ion motion has no effect on the absorption of the ordinary

ray. For the extraordinary ray, it at first increases and then

decreases the absorption. The ion motion seems to have appreciable effect for transverse propagation. The effect may well be

confined within a narrow cone.

Case (ii)Y=v ^V (lower hybrid frequency) Figures 4-7.

The behavior of the propagation factors for the longitudinal case

is as in the magneto- ionic theory. But for transverse propagation,

collisions not only remove the cutoff but reverse the trend of the

96
p

real refractive index curve (Figs. 6 and 7) for the lower

sign (ordinary ray of the magneto-ionic theory). How the

introduction of ion motion makes possible all directions of

propagation for this case has been discussed in the text.

Case (iii) "Y^ */y (Lon gyro-resonance)

Longitudinal propagation C^-o) . We shall discuss

in the text how for longitudinal propagation the ion- gyro

resonance introduces a singularity in the lower mode. Figure

8 shows that even a small amount of collision removes this

singularity. The wave can propagate but is nevertheless

subject to high absorption. With increasing collision frequency, the absorption of the lower mode decreases, whereas that

of the upper mode increases (Fig. 9).

Transverse propagation (^tpz^TT/o). The singularity is

removed in this case and the wave can propagate. The cutoff

frequencies for the two modes for no collisions is discussed in

the text. The propagation factors for a high collision case

( Z = 1000) are shown in Fig. 10.

Case (iv) ^ -sl [p* (hydromagnetic m.ode)

Longitudinal propagation(<^ao) . Figures 11 and 12.

These are the retarded magneto-acoustic mode (lower sign)

97
and the oblique Alfven mode aiffie- (upper sign) of Denisse and

Delcroix [1961]. Note in Fig. 12 that the oblique Alfven wave

(upper sign) does not suffer absorption. This property will have

considerable application in the solar corona.

Transverse propagation C<^= ^2) . Figures 13 and 14.

The lower mode suffers a cutoff at "Xo: 1 , as in the magnetoionic theory, which is removed by collisions. Note that the nonabsorptivity of the Alfven mode (upper sign) persists even in transverse propagation.

We shall. now give a brief analysis of the ion gyroresonance

referred to in Case (iii) above. From (50) we have for longitudinal

1. propagation (<^=o): C^) = (Ai,tC)/D. Taking the m.inus sign and

Z = (no collisions), we have (10?*= "n = 1 -X[ Cl + V) +^C'-')0] ,

where Y, = Si/ico • There is therefore a singiilarity in T\ at the

ion gyroresonance (Vi=l). It can be shown that a finite Z will

remove the singularity. For transverse propagation ( ^= ^^ ), we have Cnf ^ (A+BiB)/(D+E), Taking the + s^gn, we get after a

little algebra (for Z = o, 1, e, , no collisions): - » ^'^'\i--y'-Xi-Y,'';-CH-»)(i-Vi)X

When Vi = 1 (ion-gyroresonance), T\—>2-f^X • Thus Tt -^ ^ ,

as X~->0 . It can be shown that with collisions Ti -^ j,

as X-*0 . (See Fig. 10).

98
VII. Conclusion

In the present paper we have given a microscopic hydromagneto -ionic theory, to Chapman and Cowling's second

approximation, of a completely ionized hydrogen plasma. As

a special case for no collisions, i. e.^ V-Ot our formula (50)

reduces to the expressions derived by Astrom [1951] for the

longitudinal and transverse components. Furthermore the

most general dispersion formula derived rigorously from

Delcroix and Denisse's general conditions (eq. (7), p. 24, loc.

cit. ) gives the same transverse and longitudinal modes of

propagation as those obtained from our formula. As already

stated in the introduction ysiki^^^^o^t^. we have further agreements

with the results obtained by Kantor (19 63) in the limit of no

collisions. All these provide an indirect check to the consistency of our theory and correctness of our derivations.

We note that the treatment can easily be extended to a

neutral plasma of any degree of ionization. In its present

formulation, it should have important applications to hydromagnetic wave dissipation in the solar corona. The treatment

of collisions in our hydromagneto- ionic theory is broad enough

99
to include any velocity dependence of the collision cross- sections. For applications to hydromagnetic wave dissipation in

the terrestrial ionosphere, our treatment should be extended

to a partially ionized gas. Burgers' formalism should be adequate for this extended treatment, as is evidenced by Pipkin's

work [1961] on the d.c. conductivity of a partially ionized gas.

The numerical applications that have been given are merely

illustrative and by no means exhaustive of the results that can

be obtained from the theory.

Acknowledgments

It is a pleasure to acknowledge our indebtedness to

Drs. Pfister, Poeverlein, Kantor, Prasad, and Lt. Finn for

fruitful discussions, and to Mr. Arnold Shickman for the computations.

100
References

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Akasofu, S., 1960, J. Atm. Terr. Phys., J^, 160.

Allis, W. P., 1956, Handbuch der Physik (Berlin: Springer Verlag) , 21 , 383.

Allis, W. P., and Delcroix, J. L., 1963, Proc. of Paris Summer School on Plasma Physics, Vol. 1, p. 45.

Sstrom, E., 1951, Arkiv for Fysik, 2, 443.

Budden, K. G., 1961, "Radio Waves in the Ionosphere," Camb. Univ. Press,

p. 82.

Burgers, J. M., 1958, Technical Notes BN-124 a,b, Institute for Fluid

Dynamics and Applied Mathematics.

Chapman, S., and Cowling, T. G., 1958, The Mathematical Theory of NonUniform Gases, 2. ed. (Cambridge: The University Press), p. 56.

Denisse, J. F., and Delcroix, J. L., 1961, Theorie des ondes dans les

Plasmas (Paris: Dunod) , pp. 22, 85.

Dessler, A. J., 1959, J. Geophys. Res., 64, 397.

Dungey, J. W., 1951, Nature, _167, 1029.

Fejer, J. A., 1960, J. Atm. Terr. Phys., J^, 135.

Francis, W. E., Dessler, A. J., and Karplus, R., 1960, Bull. Am. Phys.

Soc. , 5^, 316.

Gershman, B. N., Ginzburg, V. L., and Denisov, N. G., 1957, Uspekhi Fiz.

Nauk, ^, 561 (AEC-tr-3493) .

Hines, C. 0., 1953, Proc. Cambr. Phil. Soc, 49, 299.

Hines, C. C, 1957, Jour. Atmos. Terrest . Phys., _11, p. 37.

Hulst, H. C. van de, 1953, The Sun, ed. G. P. Kulper (Chicago: Univ. of

Chicago Press), Chapt. 5.

Kantor, G., Whist ler-hydromagne tic Extension of Magneto-ionic Theory,

Ph.D. Thesis, Cornell Univ. (July 1963).

Lehnert, B., 1959, Nuovo Cimento Suppl., 13, 59.

101
Leinbach, H., 1962, Scientific Report No. 3, Geophysical Institute, Univ.

of Alaska, UAG-R127, Chapt . III.

Marshall, W., 1957, A.E.R.E. Reports T/R 2247, 2352, 2419, Harwell.

Menzel, D. H., 1961, Mathematical Phys. (New York, Dover Publications,

Inc.) , p. 325.

Mitra, S. K., 1952, The Upper Atmosphere (Calcutta: Royal Asiatic Soc.

of Bengal) , p. 628.

Osterbrock, D. E., 1961, Ap. J., 224, 347.

Piddington, J. H., 1956, M. N., _116, 314.

Pipkin, A. C, 1961, Phys. of Fluids, 4, 154.

Rosenbluth, M. N., and Kaufman, A. N., 1958, Phys. Rev. 109 , 1.

Schliiter, A., 1950, Zeitsch. f. Naturforsch. , ^, 72.

Schliiter, A., 1951, Zeitsch. f. Naturforsch., j6a, 73.

Sen, H. K., and Wyller, A. A., 1960, J. Geophys . Res., 65, 3938, 3942-

3943.

Spitzer, L., 1962, Phys. of Fully Ionized Gases (New York: Interscience

Publishers, Inc.) 2 ed., p. 138.

Wyller, A. A., Proc. Fifth Internat. Conf. on Ionization Phenomena in

Gases, Munich (Amsterdam: North-Holland Publ. Co.), 1961, p. 940.

Wyller, A. A., 1963a, Astrophys. Norv. , VIII, 53.

Wyller, A. A., 1963b, Astrophys. Norv., VIII, 79.

102 i
\ w \\ 1 1 \\ 1 \ \ \\ (0 1' \ Vy \\ ^ ro 1 Vi \\ oo 'l \\ \ . m^ 1 1. \ \\ • \\ \^ n 1 l' A ^ X 1 '1 \ ''x^ lO o

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103
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116
A Solution Method for Cold Plasma

J. J. Gibbons and R. E. Hartle

The Pennsylvania State UniversityIonosphere Research Laboratory

University Park, Pennsylvania

The work we will discuss stems from ideas presented in an

unpublished scientific report by J. J. Gibbonsr We will consider here an

initially neutral plasma cloud set into motion in the presence of a magnetic

field. The physical description is idealized to that of the cold plasma

approxinaation for positive and negative fluids. The cold plasma is described

by the usual fourteen dependent variables through Maxwell's equations and the

dynamical and continuity equations for both positive and negative fluids as

follows:

V. E = 4 IT (p - n)

V. rf = O

Vx H = -^ (p"^ - n"^) + -^ -^Y

^ + (w • V) w = - N{t + -^ ^ x H)

V (nv^ +|3 = O

I.

117
The solution method we have studied consists of expanding

the dependent variables in a power series of time. To satisfy the initial

conditions of a neutral plasma immersed in a magnetic field H (t) with

both positive and negative fluids moving at a constant velocity S in the

absence of an electric field, the power series takes the form

p(i^, t) = A (r) + p (l^t + p (t)t^ +

O 1 2

n(r, t) = A (f) + n {f)t + n (f)t^ +

v(?. t) = "S^ + V (r)t + V (T)t^ + ^

^ 11. w(r, t) = S^ + w (r)t + ^i^ (r)t2 +

E(r; t) = E {t)t + t (f)t^ +

1 2

H(r, t) = H (?) + H (^t + f? (f)t^ +

O 1 2

The spatially dependent coefficients are solved for by inserting equations

II into I and equating each coefficient, of every power of t, to zero. Every

coefficient is then expressed in terms of the initial values A (r), H (r) and

-J. SoWe now consider only one dependent variable, since upon

obtaining its solution the other dependent variables can automatically be

deternained. For no particular reason we choose to study the positive

fluid density p(t^, t). It can be shown that the terms of p(r^t) can be grouped

into various sets each of which is identically a three dimensional Taylor

series expansion of a representative functional of the initial values H {f),

A^fr) and S . Thus, when the above mentioned sets are summed, the first

few terms of p(i^, t) can be written as

118
t p(tt) = A^(t-S;t)+V.| dt'[^{A^Z^x^^}f^{-A wx(^xS )-2(S -VA >*xS

o ^ o' o o' o ® O ®

.A^(S^.V)cox.Sj^^+ ^ j A^w-x [ux(arxS^)]+ 3(S^- VA^)wx (wxS^)

+ 3 "S •V(S -VA) rtxS + 3(S' • VA )(S "7)5 xlp + 2A (coxS •V)Tox^

o o J o o o o o o o c O ® (D

+ A (S • V) o o

(S •V)SrxS

o o + A (S^'V)! cox (ox's ) o o

- 4Tr (P + N) A^ ojxS

o o^ O

+ A to X

o

(S .V) wxS

o ' o ® ®

jvhere the integrand variables are

®

+ 3V- (A coxS )coxS ^ + . . . ^ o o o ' in.

T = t - t' ' A = A (f^- S t') and

o o^ o ' -^ H (r - S t') C o^ o '

We know that if the power series is conditionally convergent then the sum will

not be unique for various arrangements of the terms. Therefore, for our

purposes, we will assume absolute convergence.

The first term in III represents the undeformed positive fluid

density travelling with its initial velocity S as if no force fields had been

encountered. The deformation of the distribution is described through the

integral terms which depend upon the entire past history of the plasma.

Solution III will be of definite use if one can discover initial

functions H (¥') and Aq(I^) such that the integral terms can be summed. This

would yield a model from which physical intuition can be gained and perturbation to other situations might be possible. At this date we have not discovered

119
such initial conditions. We notice that solution III can be written in the form

t Pitt) = p(r"-^^t,0)+ I

f(I^-S^t', t - f) df IV

where f(r' - S t', t - t') represents the integrand functions and indicates more

clearly the functional relationship of the independent variables. It can be

shown that all of the dependent variables can be written in this form. This

representation, which we are presently investigating, may be a useful

starting point for analysis and may be compared to the expressions obtained

by RosenblutnT

We will now point out several interesting properties born out by

equation III. First notice tkat the various sets of terms indicated by

numerals respectively form infinite series which can be immediately summed

with the result

1

p(r,t) = A^{t- S^t)

o' o '

I o' o M JJ

A (T-S t)V+-|VA (r - S t)

a>x(oj'xS )

^ t P XCjoI o

+ ') cos (toj) - 1

CO X (wx S ^ ! -'

dt'(S

o >°|SJ|7^

(S •V)S xto o o

ojxS

sin { Tw)

3!

l^xSj

(^ 'V)-\(S -V)^ x^

+ 4tt{P + N) a S xw + 2(S xw.V)S xZ^+ (S ' Vjo^x (J*x S ) o o o o o o 4

+ v. \ dt

G

w X |(So- V)-^^^^ J

+ . . . A V.

°|SJI^

^

>xS

sin(Ta))

ixS

+ (cos(tco)- 1)

ojx (ojxS )

o^

jcjx (wxS }\

120
This process can probably be continued indefinitely as suits ones purpose^and

each remaining term of a given hierarchy becomes more informative than its

predecessors.

Approximate solutions can obtain, depending upon the situation

on hand, through the neglect of appropriate ternas. For example, if we have

a very tenuous plasma and choose to neglect the self-consistent nature of

the above description, we may choose to neglect the interaction terms. In

thi s case the dynamical equations become

Q-*- v X H

^T + nr- ^ V = p —^

VI.

r. -^ W X H

-^-T- + (w V) w = - N g

For this approximation it is found that the terms with the coefficients (P + N)

are eliminated. This term obviously indicates the effect of the presence of

the negative fluid upon the positive fluid. Similar approximations may be

made when the applied magnetic fluid H is weak or strong.

Another property, evident through direct observation of equation

in, is; if the initial distribution A (r) and the applied magnetic field HQ(r)

are both dependent upon only one or two spatial variables, then at all future

times the plasma will be dependent only upon one or two spatial variables _, respectively.

As a brief exannple of the latter property let us consider a

tenuous plasma immersed in a strong but decaying magnetic field

-^ -t/T ^ H = H e ^' -^ k VII. o

In this event we assume the non-interaction equations VI are valid. If the

121
initial distribution is assumed to be a one dimensional Gaussian

A = Ae

o VIII

9 P , 9 P _ o

then, employing the above property, the continuity equation takes the form

IX.

The solution of IX is immediately

.[x-g(t)] Vl^

p = Ae X

where

g(t) = s. sin

PHT (^-f/T _ ^^ dt' ,

The density asymptotically becomes

XI

as is intuitively expected. Of course a solution for a three dimensional

Gaussian is similarily immediate.

The above approach has been useful in studying the dynannics of

a plasma cloud covering half of space and travelling toward a magnetic dipole.

Here we are interested in temporal values immediately following initiation

to correspond to the sudden comnnencement of a magnetic storm. We thus

need to consider only the first few terms in the power series.

/3] The geometry for this problem is shown in figure 1,'--' The cloud

boundary is initially a plane parallel to the x - z plane and approaches a magnetic dipole, located at the origin, fromi a point on the negative y axis

toward the origin with constant velocity S . The magnetic dipole is

pointing in the negative z direction.

122
To lowest order, the total charge density at the equator is

^ 8 A a H^

1 r. P2-N2 1 „z o

p = ^ S 21. ] t'+. . . " V. ) H^ ft " + A ft"'" } t^ +. . . XII. ^ o oyC /ozoy ooyj

Examination of the terms indicates that there is a positive charge density

everywhere near the surface with a maximum at x = o. Behind the surface

we have a negative volume charge which satisfies the requirement of charge

conservation.

The velocities to second order are

V = s" + Z. s X H t + I- — (S"- V) s" xH --^ H x(s" xH ) o C o o ZC o o o _p2 o o o

t^ + . . .

w = S '- S xHt+iU{S-V)SxH -— H x(S xH)t2+... o C o o ZC o o o ~„2 o o o

J xm.

The first order terms show that positive charges are deflected to the right

and negative to the left. The second ternn of the second order coefficient for

both positive and negative is a velocity component in the negative y direction

with a maximum value at x = o. This indicates that the oncoming positive

and negative fluids are slowed up in such a manner that the cloud front bends

about the origin. The remaining term modifies the x displacement of the

first order term. When we consider velocities away from the equatorial

plane a current pattern as indicated in figure" Z is observed on the cloud

surface.

The electric field to second order is ^ _ S^ X H

E = -ZttA (P + N) -2_«—2. t^ + . . . XIV.

o C

The generated electric field is opposite to the current, as usual in a generator of e.m. f. This clearly indicates the error in employing Ohms

law.

123
FIGURE 1

Y

If MAGNETIC POLE

>X

>oy

PLANE OF PAPER IS PLANE OF

MAGNETIC EQUATOR

CLOUD

/^

/

Ho UP IN WHOLE PLANE

» (SoXHo)

-7

dA, (NEGATIVE)

FIGURE a

''"^.

124
Acknowledgements

This work was supported by the Air Force Cambridge Research

Laboratories, Office ci Aei'ospace Research, Bedford, Massachusetts

under Contract AF19{628)-3842 and, in part, by the National Science

Foundation under Grant G~ 18983.

References

1. J. J. Gibbons, Investigation of Initial Transcient Behavior of the Totally

Neutral Ion Cloud, Ionosphere Research Laboratory Scientific Report 30,

The Pennsylvania State University, 1955.

2. Rolf K. M. Landshoff, The Plasma, in a Magnetic Field, Stanford

University Press, Stanford, California, I960.

3. The figures appear in reference 1.

GPO 841-1 19

125



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