national Bureau of Staadaras

Library, M.W. Bldg

MAY 2 2 1964

^ecUnicuL vZote

CONFERENCE ON NON-LINEAR

PROCESSES IN THE IONOSPHERE

DECEMBER 16-17, 1963

EDITORS

DONALD H. MENZEL AND ERNEST K. SMITH, IR.

Sponsored By

Voice of America

and

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

J U. S. DEPARTMENT OF COMMERCE

{ NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS

technical *^ote 211, Volume 6

Issued April 24, 1964

CONFERENCE ON NONLINEAR PROCESSES

IN THE IONOSPHERE

DECEMBER 16-17, 1963

Sponsored by

Voice of America

and

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

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

listed or referred to in the open literature.

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

Price 60():

Volume VI

Table of Contents

Session 4. Excitation Processes.

4. 2 Use of radio transmitters to decrease D-region

electron density, Paul Molmud, Space Technology

Laboratories, Inc. , Redondo Beach, California 1

4. 5 Non-linear ionospheric two-plasma processes, Joseph

E. Rowe, Professor, University of Michigan. . . ^ . . . 89

Concluding Remarks

Ernest K. Smith, Jr. , CRPL, NBS, Boulder, Colorado. . . 113

D. T. Farley, Jr. , CRPL, NBS, Jicamarca Radar

Observatory, Lima, Peru 114

Session 4 papers continued from Volume V, Technical Note

No. 211 ,

iii

USE OF RADIO TRANSMITTERS TO IDECREASE

D-REGION ELECTRON DENSITY

Paul Molmud

TRW/Space Technology LalDoratories

Redondo Beach, California

A study is made of a method of decreasing the electron density

in the D-region. The method involves the heating of electrons by

intense electromagnetic radiation. This increases the electron

attachment rate and thereby decreases the electron density. The

relevant D-layer rate processes are reviewed and incorporated into

a detailed analysis of the influence of a pertiorbing electromagnetic

pulse upon the variation in time of the electron density. The theory

of conductivity of a weakly ionized gas is reviewed and augmented to

include dependency on field strength. Propagation constants are determined and the equations describing the nonlinear radial propagation of a powerful wave in a weakly ionized inhomogeneous medixim are derived. Numerical solutions of these equations are obtained which

describe the effects due to the vertical propagation of waves for

several power levels and frequencies through a heavily perturbed Dregion. It is found that the powerful wave modifies the electron

density distribution. The combination of high frequency and optimum

power level provides a channel of decreased electron density in the

D-region through which a wanted wave may propagate with greatly reduced

attenuation

.

* The research reported in this paper has been sponsored by the

Electronics Research Directorate of the Air Force Cambridge Research

Laboratories, Air Force Office of Aerospace Research, under contract

AF lS{6ok) -Q&kk . This paper has been extracted from the final report

issued under this contract: P. Molmud, S. Altshuler, J. H. Gardner,

"The Study of a Method to Reduce Electron Density in the Ionosphere

D-Region by Means of High-Powered Ground-Based Transmitters," Sept.

1962, Space Technology Laboratories, 8625-6006-RU-OOO, AFCRL -63-753.
I. lOT^ODUCTION

It is well known that high altitude nuclear explosions can cause interference

to communication "between ground stations operating at hf (3-30Mci/s) [Glasstone (ed.)^

1962^ DASA 1229, 1961J . Periods of unusual solar activity cause similar difficulties TDut these disturbances are usually found in the polar regions [ Agard

Ionospheric Committee, I96I |

.

The communication interference is due to increased absorption in the D-region

(50-80 km altitudes) which is caused by enhanced electron density. The electrons

are produced, in the case of the nuclear explosion, immediately and evanescently

by soft X-rays from the bomb casing and then subsequently by the more longlasting sources of 3 and / emission from the radioactive debris undergoing slow

hydrodynamic dispersal. It is also thought that neutron decay may contribute to

the ionization [ Grain and Tamarkin, I96I J . The soft X-rays are absorbed in the

D-region by the photoelectric effect and thus enhance the ionization. The P's

lose their energy in the same region by ionization.and the / 's lose their energy

at lower altitudes by Compton recoils. The naturally occurring ionospheric

disturbances are thought to be due to the deposition of energy in the lower

ionosphere by energetic protons and X-rays from the sun.

The ultimate electron density which can be achieved under the abovementioned perturbed conditions is controlled by the intensity and duration of

the source and the electron removal mechanisms. The main removal mechanism

as we shall see in Section II is provided by electron attachment to molecular
oxygen. This is quite fast. Thus, the perturlsation due to the soft X-rays

from the iDomb rapidly disappears, to be replaced "by the perturbation due to

the radiactive decay of the slovly expanding debris.

The communication interference we have spoken about may last for several

days and extend over even widely separate areas I Obayashi et alt, 19591 • Such

interference is troublesome and in emergencies may be fatal. Thus^ it has been

the subject of some study.

Among the problems which have been studied are the methods by which the enhanced electron density in the D-region may be decreased. It has been suggested

that material of high electronegativity be transported from the groxind and

released at appropriate altitudes to attach and therefore remove electrons

JArmour Res. Foundation, No. 6I-I26, I96IJ Munick and Folk, I960J

.

The author of this present study suggested in 1958 [ Molmud, 1958 J that the

new data on electron attachment coefficients to could be utilized to effect

a reduction of electron density. The data, as reported by Biondi I 1958

J showed that the attachment coefficient was monotonic, and increased from at

least .OU ev to --^ .1 ev where it maximized. In addition, the maximum in the

coefficient was about twice that of its value at .Ok ev. It was reasoned^

therefore, that if the electrons in the D-layer were maintained at .1 ev

average energy then the attachment rate for these electrons wotiLd be coramensurately higher than for the unpertxirbed electrons and so the observable

steady state free electron density would be lower. The electrons were to

be maintained at this abnormally hi^ energy by feeding in energy electromagnetically via powerful radio waves broadcast from the ground. The fact
that electrons in the ionosphere can he seriously perturbed in energy by

means of powerful ground-based broadcasting stations had already been amply

demonstrated through the phenomenon knovn as the Luxembourg effect or

cross -mod\ilation [Mitra, 1952].

The idea of altering the attachment coefficient was presented at a

symposixun held at Air Force Cambridge Research Center in 1959 I Molmud, 1959Q'J •

The remainder of this paper will be arranged as follows: Section II

will discuss the important ionospheric processes occurring in the D-region.

In Section III mathematical expressions describing the density of the most

important charged components in the D-region will be set upland analytical

solutions will be obtained for these densities following an instantaneous

change of electron energy. In Section IV we discuss the theory of conductivity of weakly ionized gases for strong electric fields. We first

obtain the coupled drift velocity and average energy of electrons for the

case of a Maxwellian gas and elastic collisions only. Then we generalize

to the case of air and by an "Ansatz" include inelastic collisions. We

obtain time -dependent solutions for velocity and average energy and set up

the expressions for the conductivity tensor. Section V is concerned with the

problems of analyzing the propagation of electromagnetic waves which are

strong enough so that they change the propagation constants of the medium they

are traversing. Included are the dependence of electron collision frequency

and of electron density on average energy. In Section VI we treat the propagation of powerful waves of several frequencies and power levels through a

severely disturbed D-region. We show how these waves are attenuated and
how the attenuation depends on the power level and frequency. We also see

how the perturbation these waves make in the ionosphere facilitates the passage

of a wanted wave.

II. THE IMPORTANT IONOSPHERIC PROCESSES

The processes we shall mention have "been discussed already in several

review articles I Nicolet and Aikin, I96O; Grain, 196I; Anderson, 196I; Nawrocki

and Papa, I96I; DASA 1229^ 19^1 J

. The values presented here are taken from

these sources

.

A. Sources of Ionization

Several different processes contribute to ionization in the normal

D-region (5O-85 km). Above 70 km ionization of NO by Lyman-a appears to predominate . Below about 70 km cosmic-ray ionization of air begins to become important

and below 60 km ionization by keV X-rays ( X ** 1 A) and cosmic rays predominates.

Normally the electron density ranges from '^10 cm at 60 km to ^^ 10 at 75 km.

Under disturbed solar conditions I Fig. h of Nicolet and Aikin, I960 J , the

electron density may increase by a factor of 10 at these altitudes. Reid 1961]

calculates electron densities produced by solar protons during polar cap absorption events as even greater. Kane[l96lJ, i-eporting on rocket measurements of

3 -3

D-layer absorption and phase shift during a polar blackout, finds 2 x 10 cm

3 -3

at '^ 57 km and 6 x 10-^ cm "^ at '—70 km.

High altitude nuclear explosions contribute to the D-layer ionization

[dASA, I96I; Slasstone (ed) , I962J by (a) evanescent soft X-rays and (b) longlasting P and /rays from radioacti-ve decay of booib delJris- The ' 's deposit

their energy at about 30 km altitude and the P's at about 70 km. The maxiraim

electron densities due to the debris may be as large as lO-'-lO cm" [ riA.SA,196lJ

.
B. Charge Removal and Exchange Machanlsme

We Include electron attacbnent as a charge removal mechanlBm

since It effectively removes electrons trca interacting with electromagnetic

radiation.

1. Electron Attachment

This is described by the process:

^*°z*°z °Z*'>2

vith a rate given by K(u)[o^J n, vhere K(u) is the attachment coefficient,

a flinction of u the average electron energy, [ 0- J is the numerical density

of Og molecules esd n the electron density. The functional dependence of

K on u is given in Fig. 1, below, and is obtained from Chanln, Phelps, and

Biondi [ 1959

J

• Nicolet and Aikin [ i960] estimate that K at D-layer temperatures

has the value '^ I.5 x lo" cm /sec. _ IPX 10

8

-30

9

o^ 8 e 7

Z . uj 4

y

u_ 3

LUo

u

oCD

UJX

0.01

T—

I

nrr T 1 1—

I—I I I i _

J—I—I I I I I J I I I 0.05 0.10

AVERAGE ELECTRON ENERGY (EV)

0.50 1.0

Figure 1. Three-Body Attachment Coefficient vb Average

Electron Energy for Oxygen at 300°k [ Chanin,

Phelps, Blond 1^19 59].
2 . Photodeta chment

This is represented by:

Og + hv > ^2 "^ ®

Based upon the work of Burch, Smith and Branscombll958j the rate coefficient

is given I Nicolet and Aikin, 1960J as R = .U^i sec" with a detachment rate of

Rn , where n is the density.

3. Collisional Detachment

This is represented by:

+ X —>0 + X + e

The rate, estimated by Grain I I961 1 is essentially zero whereas Reid [196I

J

-17 3 suggests a detachment coefficient of -^-1.5 x 10 cm /sec.

k. Electron Recombination

a. e + O2'^ —>> + ,

with a recombination coefficient a =» 3-8 x lO" cm /sec.

According to Anderson

_ 1 as T '^

1961] this coefficient may depend on electron temperature

b. e + Ng^^ N + N . a = 5.9 X 10''cmVsec.

The magnitude of the coefficients of both (a) and (b) were reported by

Kasner, Rogers, and Blondl [1961J .

c . e + NO"^—> N +

-9 3/

a = 3 X 10 ^cmvBec.

[ cf. Nicolet and Aikin, i960]

I
5. Charge Exchange

a. <*°z °2*''2

a = 3 X lo" cm /sec.

[pite et al,p.96l]

At 60 km with [ OgJ = 1.^ x 10 ''cm" the charge exchange rate is J4-.5 x 10 sec' .

This is so rapid that one doubts \rtiether N^ plays any important role at these

Og + NO -^ Og + NO"*"

1961 give

i960 give

altitudes

.

h.

Nawrockl and Papa

Nicolet and Aiken

[no

J

Jtr 10 cm'^ at 60 km so

which indicates a very slow charge exchange rate.

6. Ion Recombination

0^ + 0„

a = 10 cm-'/sec.

a[no] = ._-411 -1

10 sec ,

a.

b.

c. 0" + HO'' Og + NO

Grain has suggested a coefficient of 1 x lO' 'cm^/sec for the above (a,b,and c)

7. Dissociative Attachment

e + —> 0" +

See Fig. 2, below Nawrockl nnd Papa,196l

_ p. 3 - 13 . This process becomes important for electrons only above about 1 ev in energy.

Thus, some of the electrons produced by ionizing events may be captured by

* Private communication

8
this procesSjbut these electrons decay in energy so rapidly that very few

can be removed in this way.

ELECTRON 3— ATTACHMENT (e + OJ

^

OcM

o o

CO S2

I o

CO ""

C/) X

o o

q:

u

HARRISON

and GEBALLE

/BURCH and!

/ GEBALLE

DOEHRING

1 i

2 4 6

ELECTRON ENERGY (e.v.)

8

Figure 2. Dissociative Attachment Cross

Sections vs Electron Energy
III. AN ANALYTICAL MODEL OF THE D-REGION *

A. The Differential Equations

We now set up the differential equations describing the concentration

of the important charged species In the D-reglon.

Symbols :

n electron density

n 0~ density

n positive Ion density

[OpJ particle density of molecular oxygen

q source strength of Ion pairs

u average electron energy

o subscript referring to unperturbed state

K attachment coefficient ^[e--., -HO^ + ««Og-^O^ ^ »,« + oJ

R photodetachment coefficient [ o" + h\> —^ ^2 "*" ®

J D collislonal detachment coefficient [o' +X—^Og +e +xj

a electron recombination coefficient [e + X, —> X + X

J a 0„ recombination coefficient I 0„ + X^ '^ ^o ' "2

g = q + (R + D)n_ - k[oJ2 n - a^ nn^ (III-l)

5^= ^OgJ^n - (R + D) n_ - a_n_n_^ (ill- 2)

n = n + n (III-3)

vhere Eq. (III-3) arises from requirements of local charge neutrality.

* A slightly less general discussion of the same matters in this Section

may be found in Molmud (1963)

.

10
Some steady state relationships arising fi-om Eqs. (lll-l), (III-2),

(III-3) axe as follows:

R + D + a n ^+ a_n^ + R + D + K

K103I2

^+ a_n^ + R + D + KfOgl'

(in-i*)

(III.5)

-(a^ - a_)n + y n^COg - a_)^ + 4qa_

n^ = -^

V-^ ' (^^^-^)

q + (R + D) n

n = r-^rg (III-7)

a^n^ + R + D + k[02J

q = n I a n + a_n J

• ( III-8)

The first three equations allov, in steady state. Independent solutions

for each species n, n _,and n from cuhic equations. For example, the cubic

equation for n has the following form:

2 n

+ a +

(a - a )(R + D) [or_n^+R + D-.K[o2]2]

- q

-=-^^^ 7 ^9 = ' (III-9)

a^n_^ + (R + D) + k[02]^J agn_^ + (R + D) + k[02]

We shall not concern ourselves with the general solutions of (III-9) hut the

approximate one where a n and a n can he neglected with respect to

R + D + k[o.J . Under these circumstances, n has the solution:

n
R + D + Kfo.]^ ^^^ (III-IO)

a^(R + D) +k[02]x

Equation (III-IO) can be combined with (III-U) and (III-^) to yield approximate solutions for n and n also.

B. Analytical Solutions of the Differential Equations

We seek solutions of (lll-l), (111-2), and (III-3) for the following

initial conditions: for t •« 0, u = u ,and steady state conditions prevail

and for t ^ 0^ u » u. . This is the situation "vdilch voxild obtain if a poverful radio beam vere suddenly switched on and propagated through the D-reglon.

^e energy of the electrons wovild rise almost instantaneously (for details

see Section IV) and the electron density would change ccraparatively slowly.

Case 1. a^ = a = a e

We add (lll-l) and (III-2) and make use of (III-3), thus obtaining

dn

-g^ = q. - (m^ (lll-n)

Let us assume, for the moment, that a is independent of u. Then

because of the initial conditions (i.e., dn /dt = at t = O) , the only compatible solution of (ill -11) is that n is a constant.

+ -1/2 Now let a have some dependence on energy (i.e.. as T ' ) (see

Section II) so that the solution of (lll-ll) is:

n

+00 n \ / n + n \ -2n V a ce,' t

+ -KX> +o +o ' oT. n + n M n - n

+00 +/ \ 400 +o

(III-12)

•^ere n =~\ /

^

and n

+O0 y OL +ovo Vvc^a_

12
We see that the relaxation time, t, is l/(2n-VaCL ). Let us take

rO O X n = 10 can (a severely perturbed condition), a = 5 x lO" cm /sec,

a. = 5/V2 X 10" cm-^/sec (i.e. u-/u = 2 and a -^ T

" *) . Dien t = 12 seconds

<

with corresponding higher times at lower n densities.

Let us provisionally assume that the relaxation time for electrons is

much less than that for the positive ions. Then we can solve (lll-l), (lII-2)_,

and (III-3) under the assumption that n is constant while n is undergoing

its substantial rapid changes. The solution is reaaily obtained:

-A,t -A,t

n = n^e "^ + n^ (1 - e

"^ ) (III-13)

q + (R + D)n q + (R + D)n^

where n = r , n = -t—

-— o XI 00 At_

and A^ = R + D + a^^ n^ + K(u^) [ 0^\

A^ = R + D + a ^ n^ + K(u^) I 0„J^ . eo +

Here the relaxation time t = l/A. . Its value at 60 km is obtained from

H = .kh sec"^, a^^ = 5/V?x lO""^, Kin^) [o^Y = 6.6 (i.e., u/u^ = 2),

D ~ 0, n = lO^cm •^.

Then t = 1/7 sec. At 70 km,K(u^) [o^r = 1-26 and t = . 58 sec; at 80 km,

K(a, ) L 0„J = .06 and t ^2 sec. Thus our assumption is correct for altitudes

of Importance and n can be treated as constant while n decays to n . The

subsequent time history of n is controlled by n through (III-12) and (III-U).

13
Case 2 a ?^ a

Here we shall again assume that n Is constant vhUe n imdergoes Its

major rapid variations. Then the solution for n is the same as (III-I3).

Hie differential equation for n is obtained by adding (lll-l) to (ill- 2)

inserting n for n, and using (ill- 5) and (III-7)

R + D

°el -T- ^ «-l

K(Vl02]

K '^ K^ ^(«-l- %i)

(III-IU)

Ignoring the contributions of a ,n and (a , - a ,)n to A, ,we can readily

find the solution to (lII-lU).

"+ - 'Voo n + b/c + n

-roo.

n + b/c + n \ (b + 2cn )t

' +0 ' +00 ^ +00'

= e n - n

40 +00

Where b = - QPt^^/X^, c = - l/A^ a^^{n + D) + a_^K(u^) [o^Y

and n^^ is obtained from (111-114^) by setting dn /dt = 0, I.e.,

wo

b

2c

V^

2c

kqc

(III-I5)

Here the relaxation time is given by

T = b + 2cn

+00

When, for example, at 60 lan,n = 2 x 10 cm"-^, a = 5 x 10''cm^/sec,

a_ = lO-'^craVsec, K{u^)[o^Y/i^+T)) = 7-5, K(u^) [o2]^/(R+D) = 15, then by th«

14
previous relationships, q = 3.5 x lO^cm^/sec, n = 1.5 x 10 ^cm"^,

n^^ = 2.25 X lO^cm"-^, A^ = 7 sec"''-, b»J - 3 x 10 sec""^,

-2 -1

2 en. _^ = - 5 X 10 sec . Thus t = 20 sec. For lower electron densities +00 n becomes lower and t larger. For hi^er altitudes b;^-qa /(R + D)

c i5i.a ,/(R + D) andT;::?!* sec. Ihus, the relaxation time for positlye ions

appears to be sufficiently large that, for all significant edtitudes and

fairly l€u:ge perturbations, the assumption that n is constant while n is

making its rapid change is warranted. The product an is .1 and there is

little error in neglecting it in comparison vith R + D + KLOgJ = ?. At

higher altitudes, the last sum may have the value .Mf so an error of '^20^

may be incurred —> not significant enou^ to concern ourselves with.

One can readily show that if the energy dependence of the a's is

negligible then n _/n =» 1 for a =» a and n _/n <: 1 for a <: a . +00 +o e - +00' +o e - Thus, with reference to (III-13) ^en a ^ a_, for small values of t^the

electron density decreases until n ^^n^, then n. starts to increase and. so

00 +

does n. IBius, we get a minimum in the electron density as a function of time.

Ibis minixmmi should take several seconds to manifest itself and may be

difficult to observe because of ionospheric perturbations.

The equations (lll-l), (lTI-2),and (III-3) have been programmed and

solutions for various values of q. Of , Ci ;and K(u)I I /(R + D) have been

obtained. We now examine one of the cases to see how valid our approxi-

-1 -"^ -7 ^ -1

mations are. We take q = .kh sec cm , a = 3'8 x 10 cm sec , a_ = 3.8 X 10 cm^sec"-"-, with initially K(u^)[02J^/(R + D) = 1/2 and then

K(u^)[o J /(R + D) = 1. The machine (precise) values for n and n are

15
2-3 3-3 n = 8.59 X 10 cm and n = I.289 x 10 cm . The approximate values are 3-3 3-3 n = 8.55 X 10 cm , n = 1.28 x 10 cm . The machine value of n at

o ^^ ' +0

2 -3 2 -3

t = 2.28 sec is 6.736 X 10 cm -'. The approximate value is 6.72 x 10 cm "^^

2 -3 n has a minimum at 9*3 seconds vith a value of 6.k^ x 10 cm . ^is

2 -3

minimum is quite broad. For example, at t = U.55 sec, n = 6.kQ6 x 10 cm

2 -3

and at t = 22.5 sec, n = 6.k35 x 10 cm . The approximate value of

3 -3 n is 6.k3 X 10 cm , close to that which is observed, t _ . = l.lU sec,

OD ' ' electrons ' 3 3

T,

ions

= 1.6 X 10 sec. n._^»

+00

l.k^ x 10 and n_00 corresponding *^ ** to n

+00_ 2-3 3-3

is 7<.2 X 10 cm •^. At t = 27 sec the approximate n is I.31 x 10"^cm -^

3 -3

and the machine value is 1.33 x 10 cm .

The approximate solutions ve have given appear to adequately satisfy

the differential equations.

16
IV. THEORY OF COITOUCTIVITY OF WEAKLY IONIZED GASES

In this Section, we develop the fundamental equations and relationships

for conductivity and electron average energy used in the succeeding

sections.

A. The Moment Equation

The standard procedure for obtaining expressions for average drift

velocity and average energy of electrons in a gas in an electric field is to

solve Boltzmann's transport equation I All is, 1956; Chapman and Cowling, 1939;

Margenau, 19^6 J. The method involves approximations and does not yield transient

solutions nor solutions for arbitrarily large fields. Allis i 1956] has pointed

out that the moment equation, derived from Boltzmann's equations, can be solved

exactly for the moments under certain circumstances without knowing the distribution function satisfying the transport equation. Surprisingly enough, this

procedure has received very little attention even though the resulting solutions

provide very useful information for realistic cases. We shall adopt this

procedure for the present and finally, when we can make no more progress, we shall

resort to solutions of the transport equation.

We start with the conservation equation on X(v) where X(v) is any

quantity associated with the electrons in the gaseous medium. The bar means

averaging over the distribution function of the electrons. We shall assume

spatial homogeneity in what follows:

^

(nX) - nV^ '[^^ + 0^ X 7] X -J(X' - X)F(V)f(v) | v-v|a( e,| v-v|)

. d^d^vd-^V (lV-1)

17
where n is the density of electrons, ax = - .*- q, = -|e |, X* - X is the

change in X due to collision > F(V) is the distribution function for neutral

particles, f(v) is the distribution function for electrons, (t( 9,|V-v|)

is the differential cross section for scattering an electron of relative

velocity v-V through an angle into a solid angle d W . The integration

is over solid angle and velocity space. V * means divergence operator in

velocity coordinates.

1 Let X = V

Then, assuming elastic collisions JAllis, 1956, p. ^23 I

^(nv^) - n(^ E +^xv^) = jj-^^ J

F(V)f(v) Q( | v-vl)(v-V) I v-V I d^vd^V (lV-2)

\rtiere Q(Iv-v|) =/ (l - cos e)ff(e, | v-V|)d W , the cross section for

momentiun transfer; M is the mass of the molecule, m the mass of the electron

and v = / vfd V, the average drift velocity of the electron.

2. Let X = |mv^

Then

^v* m ^ ^'"^^ =/m ^ • ("^v i"iv^)f<i^v = qE

-J

vfd^v = qE ^d

V^- (oi^xv) ^mv = oi^xv • mv + |-mv V • {(kxv) = ^mv ( V xv • al ) = .

Proceeding as detailed in Allis [l956, p. kzk]

^ (nu) + i^-^ Kmv + mV) • (v-V)q(| v-V|) |v-V|f(v)F(V)d^vd^V = qE - v^n (lV-3)

(M + m) J

vrtiere u = ^mv .

18
3. Let Q(|v-V^|) |v-v| = constant

For the case of Q( |v-V|) |v-V| a constant (a Maxwell Ian gas)

Eq.s. (lV-2) and (lV-3) may be reduced to the convenient forms,

(M + m)

Where V = pQ(| v-V I ) I v-V

I

^d =jvF(V)d^V

%-Ji^^F(V)d^V .

In deriving Eq.. (IV- 5) from (lV-3)> it is assumed that F(V) is Maxwellian

centered about a velocity V, .

We see that (IV-^) and (IV- 5) are tvo coupled differential equations for

V, and u.

d.

The first of these equations has already been obtained by Altshuler I 1963

and others I Kihara, 1952 . The second was obtained by Molmud I 1956 J by a less

general procedure. Since they are both linear, exact solutions can be vritten down

immediately for any E(t). We postpone investigating the solutions, however,

until later.

19
V - V|) = c |v - V |, where c is a constant.

It has "been olDserved [ Molmud,1959^; Phelps, 1960 that the momentum transfer

cross section for air is most closely represented (for temperatures far from

absolute zero) as varying linearly withlv-v|. Under these circiimstances, (lV-2)

2 2 and (lV-3) (neglecting terms of order m/M and V /v ) "become:

3t m + M

9t ^.(|^^.|.„.^} =.-.., .

Again as in the case of the Maxwelllan gas^ ve have tvo coupled

differential equations, but here the coupling is stronger because now \> is

a function of the energy.

Equations (lV-6) and (lV-7) are not entirely useful as they stand for they

contain averages over the distribution function of the electrons, \rtiich is not

known. TOiey do, however, contain useful information. For example, in

Eq. (lV-6) there is a coupling term between the pressure term of the electrons,

vv*, and the average drift velocity of the neutral molecules. In Eq. (IV- 7)*

if the distribution function is assumed Maxwellian then v /v = (5/3)v .

Thus ignoring V. and assvmiing the isotropic part of the distribution function

is Maxwellian, then (lV-6) and (lV-7) become

20
^^mT¥ ^^^d = m^^^^^d (^-9)

9u , 2Mn ^, 5 r \ ;^ -" /„, «\

-^ vinv

\riiere v.k = —2 . d u

We see that (lV-8) and (lV-9) "bear a formal resemiblance to (Vf-k) and (IV- 5),

the only differences teing in the presence of k and that v> is now a function

of energy.

We here concentrate on solutions of (IV-U) and (IV- 5) and shall retvim

later to (lV-8) and (lV-9) to see how their solutions may be obtained.

B. !Ihe Solution of the Moment Equation for Drift Velocity

1. The Maxwell ian Gas

We seek solutions for Eq. (lV-4) under the following conditions: —

-» _», _i iixst for t -= 0, V = 0,' E = 0; for t >l 0,E = E e

o

..^ —

Let us call E that component of E perpendic\ilar to H and E the

component parallel to H . Furthermore, the component of the average velocity

perpendicular to H is broken up into a component v^^ , parallel to E^^ and v^,

perpendicular to E. Hei^ and in what follows, we omit the subscripts d.

The various components may be represented as in Fig. 3,

21
•P^ H

z

Figure 3. Coordinate System for Eqs. ClV-10,11,12)

The equations to be solved are now:

V + \> V =

° E

P P m p

V, Hq

V + V y, = -» E, +

M II m 1 cm

qH V +v>v =-v 1 X II mc

(lV-10)

(lV-11)

(lV-12)

The solution of (iV-ll) and of CtV-12) will provide the solution

of (lV-10) upon setting H = 0. Prom (IV-U) and (lV-12) ve readily obtain

the following:

\ +20v_^ + (^2^a>^^ ^) \ = S\% ' (lV-13)

22
the solution of vhlch Is

V = ^ 01 e / e m L^ dx dT . (TV'lh)

Upon substitution of the time dependence of E, decided upon above,

ve obtain,

V

'I "^^^o / e^ \ " m V+ i(to + oi^) S v> + l(aj - oi ) ' v^ + i(a) - oi )

(-\) + lax )t -\)f. . y

b' e sin ovt

'^

. (IV-15)

dus V becomes;

\ = i[v+ i(t°+fl^)3[v+ i^oi-cu^j) [

( ^ + 1^)

iojb

and V : P

+ ^iox e (-v^ + la^)t. _^ 1 ->^t . ^

J -e cos 01 1 \ ,

e po

E_. |e e ""] iojt - v>t

p m V + io)

(lV-16)

(lV-17)

The solutions ( IV-15) through (IV-I?) reach a steady state in a

time roughly several times l/v . The collision frequencies,"v^, we shall be

7 -1 -6

dealing with are such that V is ^—' 10 sec' and thus in lO" sec steady

state conditions for v will be reached.

The conductivity of an ionized medium is defined by setting

J = nev = cE. Since in the present case v is not In the same direction as

E, a is a tensor ^ich we shall denote by || a || . The conductivity tensor

maybe written as [Allls, 1956; Molmud, 1962aJ.

23
--k

vhere

L + R i(R - L)

i(L - R) L + R

ZP

P =

L =

R =

(lV-18)

U)

\)

1

V + 1(0)

1

+ co^)

V + 1(0) -V

m

and the z axis is chosen to be in the direction of H.

2. Air

We cannot proceed with the solution of Eq. (lV-8) in the same

manner as (iV-k) for two reasons: (a) the value of k depends on the distribution function which has not been determined yet and (b) the value of

V depends on u of Eq. (IV-T)' If we restrict ourselves to weak fields so

that u - u < e where e —^ o, then (lV-7) is decoupled from (lV-8). Additionally, reasoning by analogy with the results for the Maxwellian gas, we

expect that v will achieve steady state in approximately a time of l/k'^ .

Thus, all that need be considered is the steady state solution of v. For

this, we now turn to the transport equation.

2k
The solution of the transport equation for the distribution

function of electrons under the influence of an oscillatory electric field has

already been determined by Margenau 19k6j and Allis 1956 Margenau has shown

that even for u/^q = 2, the isotropic part of the distribution function is

Maxvellian. The resulting conductivity tensor has the same form as (IV-I8)

with some'tAiat different definitions for P, L and R. Ibese are now (isotropic

part of distribution function assumed Maxwellian)

:

P =^

^Ifl

3/2

00

ve__dv

iu) +V(v)

'''T P|«l J„ l(<o - 0^) + V (v)

m

\rtiere p = "=r=f , V (v) = pQ(v)v. lliese integrals may be evaluated, and

functional forms are displayed in Molmud Molmud [ 19 59b and Phelps [i960

[1962a, (and for an equivalent treatment, Shkarafsky, 196])J has shown how to

treat these functional forms as thou^ they arose from a Maxwellian gas. The

end result of the analysis is that V in P, L and R of (TV-I8) is to be replaced by g, where g = g„ + ig , and g and g. are functions of v/Q where

vpQfd-^v and W = o) i ca .

Thus to use (IV-I8) for air, we write

V =

25
p

1

L =

R =

g^ + i(a) +cu^ + g^)

g^ + i(cD - CUj^ + g^)

•vdiere the superscript + or - refers to taking the + or - sign

respectively in W = (o - ai • The absence of a superscript means that cu

is to be taken as zero. Figures k and 5 below [from Molmud, 1962aj respectively present values of g /v vs (cd ^ "V>)/x> ^^^ 8j/(oi * oi ) vs

C. The Solution of the Moment Equation for Average Energy

1. Maxwellian Gas

We now solve Eq. (IV- 5) (neglecting V, ) with similar initial

conditions as used for v, i.e., t'*0, E = 0, u = u; ts.0, E = E e

However, E • v in Eq. (IV- 5) is to be interpreted as meaning real part of

E dotted onto real part of v. Thus

~^ ~^ e 2 e 2 E • V = —moo A cos ojt + B sin cot cos ojt + — A cos ojt + B sin <xst cos oA , m

\rtiere the transient terms in v have been neglected, E and E. are defined

po lo

in Section IV-B, and

26
10+2

10

.+ 1

10

3+ 1 3

10

-I

10,-2

10"

Liui|.[i.ijji[.thttt1tfl''tt^^^^^

I 1 «^"vs -^-'» NiJIJl-Uilil 1 ]

1 1 1 1 1 1 1 1 mj_|_LI flr 'S '>^E REAL PART OF THE Bi^^TTl 1 11 B

= Bi==EE = i== = E^ EFFECTIVE COLLISION FREQUENCY = = = = = = = = = = = = = = m = =

EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE = EEEEEEEEEEEEEEEEEEEEEEEEEEEEEE3^EE

. iL

|4-[

1 II

t M

1 II m 1 1 1

1 [TTfltttlttttittt ll H 1 1 II I'fl r II 1 ~ T^ ' r iTTTTITir ~ ^= Jl = ^ S Tt = _[ 1 E EE E 1 1 1 [^ 1 1 1 1 1 1 |=|

1 ill!

Miinii^nni^inHiiHM^HiiiiiiiiiM|MIHiiniEiiiiinili^iilii=

^« IMn

^.M

EEEEnq?hEEEEEEEEEEEE=n EEEEEEeHeEE EEEEE^ ^ 1 1 i

I

\\M\\\\\\\\\\\\\\\\\\\\\\\\\\M m == === ======== ==== ====== = === = === = ====== ==== ===== ===== ======= ==i

J = = == = = = := = = = = = = = === = = = ==:^ = --r = =-- = = i==r= = r= = = = = ==== = = = = = === = = = = = = === = ||||||||||||||l||y||||||||||||||||||n|^

t _

lllllllllllllllliiiilllllllllllllllll[|i|||||||||||||lMi

========== ========5= =====:==:==========: =EE=EE===: ===========

: r z ::::

1.0 I! 1.2 1.3 ^ 1.4 1.5 1.6 1.7

V

Figure If . The Real Part of the Effective Collision

Frequency vs J2/v

27
10

+1

10

3

-h; 3

10"

10

h- m::i::::::::--:::::: EEEEEEi:;;; j;;;s 4^M^mllllllll 1 IlillllllllilMf+i iiiiiiiiiiiii iir [44J

i 1- —+_.... }... •i~ W±W|,

Qj* IS THE IMAGINARY PART OF THE

EFFECTIVE COLLISION FREQUENCY

eH

^^6= mc

= Trt1 1:!s;;;:::-=^:;::: ^-""::::- :: -f+ ::::::::::^^|^ -::-:^: .:::::- k r J J-L

:_s::::::::: :::::: l—Vi.llizzVzV:':':

=JL1[_1 III j' 1 1 1 ||l|{

1 1

:!;;:;;::::: -eee

-- iX-- 4 ^1

-^

S-

__ h ::.:|.:|. = :::::::

.j- J. -^ - - " -T _ _ . I , li _4,Il.. TT

Ifl-J-... MM ||N|||||H|iHll +i ^

—

---

1 :::::::: :!; ::: :::::J--(fi3:^:::: -^

I----- -- n

f'titt"

= = = ::::: :::: = = 5 .MJt,!.,,,

= = = = : yzii\\\\\'::::. i:|:::::ap::5:;

jr=p--p|-: ::::::\::zzf

1 if

:::::' J"::::::: STriTF — X

1" N :ASYMI lit '|- )R X li

= — 1

1

1 1

1

1 llllllllll f

^-

1 Ij n]

I

'^•' = =m; ^ -.: wfflik*! — ^i

- U\ —\-- - I-- t--

It ^ \ ;::::. ::l ::_;::

" "t ""t t:;

Jm t_V 1 1 :_:_: ^- + -.S. -I--.

,.. ---4; !] —1'^ "" ^--- j- 1 ;. t :: :: ::::.;i; :..: :: J J T^

t 1 ::::::::j. :--:i:; \ jj

= :::::::: :::i e=e|;2|;; i-.iii m:S:EEEE==;:

iinm:: :::::: 1 P

±__: ::::::::: :::j - r-

" _3-::::: ::: ::::,:: 4-::

E -^::E EziiEEEEEe:;; :;|:dEi !:;;:. |: IeHee;; Pill M E^i;

-_ *j:4; i^ -r -^ - —1—V- — -

r

"' :" ^"J '^i 1 . .J ... _- - - — \^ —

'

' ;

f.. I ... t. _ ^ - --

_. .

ll _2 -

1 ....

" i::::: ffil -J-Il -- i i

T 1IT "*"

> i _ . 1 ^ <- J-- T--- -- r3 it .. _J L 1... 10" 10' 10^

Figure 5. The Imaginary Part of the Effective Co3JLislon

Frequency vs f2/v

10

,^-|

28
A = -^ T-^— , ^ - "^ ' ^o

?. , Z

0) E

o 2 ^2 po ' o~ 2 2

The solution of Eq. (IV- 5) follovi

2 e o dm

i^ [A^f a] f

d-e-^'^^) + gJg^cos 2a3b^-^ 2co nln 2cat - ^e'^^H 'j (iv-19)

°

l G v) + Ikn j

e^(B + B)

2m(G^v>^ f Iku'') — [o-v^sin 2ajt - 2aj cos 2ujt + 2uje~

J ,

and G =

^ , the fractionaJ. energy lost per collision.

(M + m)'^

Now when 2a> =» Gv> only the following terms remain:

The above inequality usually holds since G?=*10" for elastic collisions and

2 Gy^^lO for regions under consideration. Thus, wave frequencies higher

2 -1 -3

than 10 sec will satisfy this inequality. Later when we take G '^lO

h -1

to account for inelastic collisions, frequencies higher than 10 sec will

be required, but this too will be satisfied for our applications, "nius, we

shall deal with the fairly simple form (lV-20) for the average electron energy.

29
A relaxation tine of l/GV '^lO" sec for elastic collisions Is

obtained vhen V = 10 sec' . When G = lO' then l/GV^^^ lO" sec. The

electron density relaxation time Is much greater than these relaxation times.

Thus, If (lV-20) Is representative for electrons in alr^then ve are Justified

In assuming In Section III that the electron energy may be changed Instan*

taneously.

Before going on to examine the solution for u for air, let us

consider (IV- 20) In more detail.

Equation (IV- 20) Is now written In s inkier foxm:

" = % "^ aiGyJ 2 »2 ^po "^^^Aor, ^2 72 "/ aiMy\^^c^^c po 72 72 -^^((D + (B^)' +V (o) - 0^) +W 1

Ve observe that under conditions where o) - odl = and \> •* (a, then

_2

1 - eGVb

•(iv-2i7

o 2mG K7

1

-G»>t

1 . (IV- 22)

allowing enormous value for u, steady state, for very small values of ^.

2 1 19

We note for future reference that the magnitude of e /2m Is 7.9 x 10 "^ ^en u Is in electron volts and E Is In stat volts (esu)

.

2. Q « c |V v|, Mr

In the case of Q varying as the velocity^ we have seen that In

the expression for drift velocity the collision frequency Is to be replaced

by an effective collision frequency, g, vhich Is complex. In addition, we

30
shall take G, being the fractional energy lost by an electron to include the

effects of inelastic collisions (vhen collisions are elastic, G has the value

2Mn/(M + m) ) G is now to be a function of u and u and of magnitude determined

by experiment. The differential equation for u takes the form suggested by (lV-9).

Il + G(u, u^) ^)(u - u^) = q E . V , (lV-i.3)

where v^= V u/u

o o

A good approximate solution to (lV-23) when a> > Gv> and also

0) ^ V , or 0) -c \) and u/u not much greater than 2 is:

o' o ' o

2 e u = u +

o 2mGV

^^ e2

(CO + g^)^ + g/ P° (IV-2U)

Coj + oj^j + g^) + g^ (to - (0^ + gj.) + gy

1 - e

We shall consider that the u of Eq. (IV-2U) can be effected

throu^ a ground-based radio transmitter. To determine the power requirements

of such a transmitter^we examine the requirements to obtain u/u^ = 2 at

various levels of the D-region \rtxen the frequency of the transmitter is at

the gyroftequency (i.e., oi = ax) and the transmitter is isotropic. We take

an experimentally determined value for G, G = 2 x lO" I Barrington and

Thrane, I962

J . The results are listed In Table IV-I.

* Barrington and Thrane obtain a value for G using an equation similar to

(lV-23) in an analysis of ciross-modulatlon experiments.

31
TABLE IV1

Ground-Based Power Requirements to Effect u/u = 2 at Cyclotron Frequencies

Altitude

60 km

70 km

80 km

ThuBf fairly large power is required for the low altitudes unless

directive antennas are employed. However, it is probably hard to get gains of

more than 10 db at cyclotron frequencies.

Powei• (watts)

21 xlO^

17 xlO^

1^.8 X 10^

32
V. NOMLIMEAR PROPAGATION IN WEAKLY IONIZED MEDIA

In this Section, we shall concern ourselves with the propagation through

an ionized mediiiin of electromagnetic waves so strong that they alter the

medium through which they propagate. Ihus we shall treat nonlinear propagation

problems

.

The important alterations of the medium will occur "because the electrons become heated, change their collision frequency and also, because of the processes

discussed previously, change their density. All these alterations depend on

the electric field strength which in turn is based on the manner in \Aich the

wave has propagated to the region under consideration. In Part A, below, we

set down expressions for the propagation constants which, since they depend

on electron density and collision frequency, also depend on the field strength.

It will be assumed that steady state conditions prevail so that the propagation

constants have no time dependence.

A. Propagation Constants for WeaJdy Ionized Media

Lorentz's equations for an ionized medium are VxH = i4f.i^nev (V-l)

VxE = -i^ (V.2)

V • E = Ujip (V-3)

V

• H = (V-lt)

33
i^ere v from Section IV, Eq. (IV-2) Is to be employed and n is a function

of average energy (Section IV, Eq. (TV-A)) and position.

Nov, by the ustial manipulation, ve form the wave equation and

assume harmonic dependence for E. TtaxB,

V2^*4(i-^)^=V(V-i) (V-5)

and Hall Is obtained fl>om Eq. (IV-I8) of Section IV. || all Is also a

function of E since a depends on v^ and n idalch are functions of average

energy of the electrons.

Plane wave solutions of (V-5) (neglecting V (V ' E) ) give

E = E e ^\^ere >^ls the propagation constant. Expressions for ^for

various directions of propagation of plane waves with respect to the

static magnetic field have been given by Sen and Wyller[l96o |, and

Molmud [1962a J . Only the case of propagation parallel to the magnetic

field is considex^d here. Then,

and from Section IV:

g^ + l(a> + 01^ + g^

(V-6)

R = g^ + 1(00 - (Ujj + g^)

34
The vave associated with L (in V-6) is the ordinary wave and is left

circularly polarized atout the direction of propagation. R is associated

with the extraordinary wave which is right circularly polarized.

If A= |i - ik where n is the real part of the refractive index

and k is the absorption coefficient, and it is assumed a> /a>(a) - m) -< •< 1,

then

(o) ± (0^ + gp

± 2

0)

(V-7)

±

[(03^0)^^+ g*) + (g^) J

Also ^en o) = oi , then:

(v-8)

^i>. (v-9)

In addition when oo = ai and V= o) then

D pi

= I V 2 ^%' • (V-10)

^^x = ^x =

c V t ^ %

As a ntimerical example, let us compute the attenuation suffered by a 7-1 3-3

gyrowave. Consider that cd = 10 sec = ax and n = 5*5 x 10 sec so that

2 2 12 -2

CD = ^ = l8 X 10 sec corresponding to 70 km altitude. Then,

It

35
This is an enormous attenuation and one \rtiich would n6t allow the extraordlnaiy

conponent of the gyrowave to reach any appreciable height in a disturbed

D-region before it is ccsnpletely absorbed. The attenuation of the ordinary

component may be obtained from Eq. (V-8) and is k = A3 nepers/km = 1.9 db/lon,

much lower than for the extraordinary wave.

We can now anticipate one of our conclusions, that trying to modify

an already distxirbed D-region by means of a wave at the gyrofrequency will not

succeed, since the extraordinary component will suffer too much attenuation

before it can alter any significant depth of the region.

B. The Eikonal Solution for a Radial Strong Wave

The field in a vacuum drops off as l/r at large distances from a

powerful transmitter. The field in a medium whose propagation characteristics

depend on the field strength will decay in a more complicated manner, but at

least as fast as l/r. We shall now seek a solution of (V-5) for a radicG.

wave. We expect the resultant solutions to be valid for conditions of little

ray bending and over small solid angles with the transmitter at the origin.

If y(V • E) is neglected, (V-5) may be written

1^ (rE) + 4 (1 - ''"^^^'^'^

) Er = . (V-ll)

Sr^ c

'^

We seek a solution of (V-ll) such that rE = A(r)e^^^^^ . Then

A(r) and (|)(r) must satisfy the following (from (V-ll) ):

36
A-A[|fr=-4A-^">V (V-X3)

^r

ar A 3r ar 2 r

' ^v-l3;

c

\daere a = a + ia. and 9 Is a function of the field strength. On the

assumption that 9 A/ 8 r «= < Acd /c ^then, if ve neglect the effects of

magnetic field

r ... 2 i

r

o

/ a> I - ^ ^

dr- . (V-14)

(o) + g^) + g^ /

Additionally, when ~ * -= 1 and 2%/ '^r^ is negligible then

(<o + g^) + g^

2

(u g

1 3A P

^

I I? f 72 21 • (V-15) ^^^ 2c [((o + g^)2 +g/]

A similar development and an examination of the sufficiency for the

approximation used have been given in an -unpublished report [Molmud, 1959c].

The following sequence of transformations puts (V-I5) into a more

useable form:

2 *

Hbus, upon writing E for EE , we obtain

2

c

[ (o) + g^) + g^ J

37
It was shown In the previous Section that, the average energy of

the electrons is a function of E (Section IV) and that the collision

ftrequency, \> , in air varies llneeurly with average energy. Since g and g. r 1 are functions of vJ/oj^then they too depend on energy and thus E . Purther2

more, a> is proportional to electron density and it was shown that, that too

2

is a function of average energy (Section III) and thus E . Therefore,

Eq. (V-l6) may "be considered as a differential equation for E . Similar

equations have "been solved analytically for homogeneous media Molmud, 1959c;

and Ginshurg and Giirevich, i960

for inhomogeneous media.

No analytic solutions have "been ohtained

In order to ohtain solutions of (V-I6), for an inhomogeneous medixam

(such as the D-layer) we must resort to programming. The results of such

programming will "be demonstrated in Section VI -vrtiere we concern ourselves

with the pro"blem of modifying the electron density in the D-region by groundbased powerful reidio transmitters.

Ihe propagation problem treated in VI concerns itself with the

propagation of a powerful wave vertically through the D-region to an altitude

2

/ ,v 2 kits

Of 100 km. The field strength is to satisfy Eq. (V-I6) and cd = —— n^

-vrtiere

n (z) R + K(u ) foj^

R + K(u) [O3]

n(z) is a chosen distribution of electron density and n (z) is the

quasi-stationary electron density due to the alteration of the average electron

38
energy u by the powerful vave (cf. Section III), K(u) is the attachment

coefficient (Section II, Fig. 1) . In addition, u is given "by the following

relationship.

7.9 eV X lO-"-^

"o Gujz) [(o> + g^)2 +g^2J

^ere u (z) is the average baclcgroiind energy of the molecules as a function

of altitude and is in electron volts, g and g. are functions of V (z)/co u/u

] given by Figs. i^. and ^ of Section IV. Inasmuch as there is no data on the

dependence of G on u and u in the region of .03 eV "« u «« .2 eV, G will be

assumed constant.

We take advantage of this last simplification by writing (V-I6) as

h^-

2 lE^ ^ 'g2

0) g

-1 ^-^4

2l ' (v-^9)

which upon combining with (V-l?) and (V-I8) results in an equation for u/u .

The wave propagation problem will be uniquely specified by supplying

03 and u(z )/u (z ). Fran (V-I8) we see that u/u is linearly dependent on

E ^ich in turn measures the power requirement of the transmitter (assuming

no attenuation of the wave up to z )

.

39
VI. PERTURBATIONS IN THE D-REGION CAUSED BY A POWERFUL WAVE LAUNCHED FROM

THE GROUND

A. Details of the Program

In this Section ve investigate the modification that can be made of

a D-region due to a powerful wave "being propagated through it. Our purpose

will be to see under what conditions and to \^at degree this modification will

facilitate the propagation of a wanted wave. We also investigate the

characteristics of the nonlinear propagation of the powerful wave.

We choose a disturbed D-layer electron density distribution which

might be associated with the explosion of a high altitude nuclear device, the

electrons being produced by deposition of energy via the 3 particles from

the radioactive debris. Figure 6, below, represents such a distribution and

ccsnes from DASA 1961 . Table VI-I gives the necessary atmospheric data.

The temperature and density were obtained from the AEDC model atmosphere

Mlnzner, Champion, and Pond, 1959 y electron density from Fig. 6 (daytime)

and collision frequency from Phelps i960

Biese data are used to compute the attenuation of a weak 30 Mc/s wave

(the wanted wave) propagated vertically from the ground to an altitude of

100 km. We then Illuminate this same region by a powerful wave of the same

frequency -vAiose i)Ower level is determined by specifying u/u , the ratio by

which electron energy is changed at 4 5 km by the presence of the wave. Fortyfive km is chosen because the wave will have suffered negligible attenuation

* P = 1.0^ X 10 p sec' where p Is pressure in ram Hg. p is obtained

from Mlnzner, Champion and Pond 1959 using geopotentlal altitudes.

** We shall assume that the average energy of the unheated electrons,

u , is the same as the background gases and have the values as given

in Table VI-1. For Justification of this see Appendix II.

ko
up to that altitude and thus the specification of u/u^ will uniquely

characterize the power level of the transmitter.

140

130

120 ^lU

»- no

LU5 100

-i^ 90

1 lUa 80

3t- 70

H-J

< 60

50

40

30

20

I\

\ LVN

^^ \

S S

NIGH'k

— l) ^^^DAY -

:::;l —

*""

10^ 10^ 10^ 10-

EQUILIBRIUM ELECTRON DENSITY - ELECTRONS/CM'^

Figure 6. Equilibrium Electron Density due to

a High Altitude Nuclear Explosion

10^

In
The powerful wave vill modify the average electron energy and

density, n lieing changed to n . It will consequently suffer a different

attenuation than the weak wave.

Now \rtien the powerful wave is turned off the electron energy

quickly returns to normal so that u/u = 1 every*rtiere, but the electron

density remains near n for a time of the order of a second. We then propagate the weak 30 Mc/s signal through this newly modified ionosphere hefore the

electron density can relax to normalcy. !Ihe attenuation of the weak 30 mc

wave will differ frcsn that of the strong 30 Mc/s wave because the collision

frequency is altered. It will also differ from the attenuation experienced

by the weak 30 Mc/s wave sent up before the poweful wave because the electron

density distribution is altered.

The abo\3 process is repeated for waves of frequencies Uo V 2, 8o

and 320 Mc/s. For each of these other frequencies we follow the intense wave

with a weak 30 Mc/s wave through the perturbed D- region to see how effectively

the other frequencies at the various power levels have "swept out" the

electrons in the D-region. We do not consider any frequencies lower than

these (e.g.4 gyrofX'equency) because of the excessive attenuation they would

experience which would render ineffectual their capability of altering

ionospheric electron densities.

k2
TABLE VI1

Parameters of the JJpper Atmocphere

Employed In Computational Prograjns

Alt .km u eV

o

.0338

V (sec

2.24 X 108

a

1.7 X

(cm-3)

10^^

n (cm'"^ nv(cm sec )

Uo

h3 .0350 1.52

rt 1.2

U5 .0358 1.22 X 10^ 8.8 X 10^5 5 X 10^ 6.1 X

10^°

hi .0366 9.50 X lo-? 6.8 X 10^5 103 9.5x10^°

50 .0365 6.55 X 10^ 4.7 6 X 10^

1 39 X 10^°

52 .0365 5.10 3.7 10^ 51 X 10^°

5»^ .0360 4.10 3.0 3 X 10

1 12 X lO-'--'-

56 .0350 3.14 2.4 4 X 10

1 13 X lO-"--*- 58 .03U0 2.42 1.9 8 X 10^ 19 X lO-*"""- 60 .0330 1.90 1.5 105 19 X lO-'-^ 62 .0320 1.43 1.2 X 10^5

_ 1 1.5 X 10^ 21 X lO-"--"- 64 .0310 1.03 jf 9.5 X

10^'' 2 X 10^ 21 X lO'^-'- 66 .0300 7.85 X 10^ 7.4 3 X 10^ 24 X lO"*""^ 68 .0290 6.00 5.7 3.3 X 10^ 20

TO .0280 4.30 4.4 3.5 X 10^ 15

72 .0270 3.24 3.3 3.8 X 10^ 12

7i^ .0260 2.36

^ 2.5 3.8 X 10^ 9.0

76 .0255 1.68 X 10« 1.8

lo^-*

3.8 X 10^ 1.5

78 .0225 1.12 1.3 X 3.8 X 10^ 1.0

80 .0214 7.75 X ,,5 9.2 X io" 3.5 X 10^

82 .0214 5.22 6.2 3.4

&k .0214 3.50 4.1 3.2

86 .0214 2.34 3.0 X 10^

88 .0214 1.57 2.5 X 10^

90 .0214 1.05

1 2 X 10^

100 .0257 1.68 X

10'' 10 5

^3
The method of computing the propagation of the strong wave was discussed

in Section V.

To recapitulate: we are assianing the fractional energy loss on collision,

G, is constant; vertical and radial propagation; collision frequency proportional

to average energy, and steady state propagation with n replacing n. The latter

assumption requires that the wave be on long enough to achieve n but not so

long that recombination effects play an Important role. It was shown in Section III

that a wave lasting •- -^ second will suffice.

K(u) will be obtained by linearly extrapolating the data of Biondi

I Chanin, Phelps, Biondi, 1959 • This gives somewhat higher values of K at D-layer

temperatures (i.e., 2.8 x 10~ cm /sec) than is estimated by Nicolet and Aiken 1 1960 .

Thus predictions of electron density reduction will be somewhat pessimistic.

We shall specify that u/u have the values 1, 2.52, U.OO and 6.00

at U5 km. Itie value u/u =2.52 was chosen because this maximizes the value

of the electron attachment coefficient, K, at U 5 km, while u/u = U.OO at

45 km maximizes the coefficient at 60 km, the altitude of maxixmjm attenuation

per \mlt length (assximlng l/r dependence of power and no absorption) . Ihe

value u/u = 6.00 Sclavs for some absorption so that the wave may have enovigh

power to reach to hljjier altitudes and alter the D-reglon.

Table VI- II present the essential results of the programs.

Uh
TABLE VI- II

Attemiatlon of Waves Sent from Ground

throu^ D-Reglon of Figure 1, Section VI

Freqtiency

Mc/e

Weak Wave

Atteniaatlon

db

Strong Wave

Attenuation

db

Weak Wave After

Strong Wave>

Attenuation

db

Weak 30 Mc/s va\^'

After Strong Wave«

Attenuation

db

u_u

= 2.52

30 8lt.77 83.1f2 78.92 78.92

iw VT 2^^.75 26.1^8 21. U6 73.93

80 12.51 Ik.kQ 10.19 69.61

320 .79 1.09 .58 62.53

30 8k,k7 85.09 78.35 78.35

koVT 2lf.75 29.i^9 21.09 12. hi

80 12.51 17.36 9.91 67.30

320 .79 1.53 .53 56.78

52.31 optimum

^= 6.00

30 Qk.kl 87.28 79.13 79.13

koiV 2U.75 32.90 21.^0 73.13

80 12.51 20.79 10.12 68.23

320 .79 2.58 .60 62.1*6

45
The most Important resxilt is this: Ttie -week 30 Mc/s (wanted) wave

propagating vertically through the D-region of Fig. 6 will

suffer an attenuation of Qk.kl db. By using the strategy of propagation

through this region of a wave of 320 Mc/s so powerfiil that it causes u/u^ = 4.00

(raises the energy of electrons by a factor of four) at 45 km altitude, we

modify the electron density distribution. Ohe 30 Mc/s wave now propagating

through this modified region will suffer an attenuation of 56.78 db; a

decrease of attenuation of 27.69 db or an increase of power level of 590.'

This decrease of attenuation is quite close to the optimum -vdiich is obtained

by keeping u at .09 ev and thus K at maximum throu^out the D-region. Tbe

optimum attenuation is 52.31 db, down by 32.16 db from, the original attenuation. Thvis the best we could possibly do by any strategy of heating the

electrons by ground-based transmitters is to increase the received power

by 1650 (see Part D of this Section, however).

B. Examination of Figures 7-17

We now present the curves resulting from the above described series

of "Gedanken" experiments. Kiese curves present the dependence of the

following on altitude:

Attenuation: defined as p

where subscript 1 refers to quantities measured a.t h^ Ism and z is altitude.

k6
^'/^ \ : ^ere u is the average energy of the electrons and u

their unperturbed energy

n (z) oo

•^rr\—: where n is electron density as obtained from Fig. 1 and Table I and

n^(z) R+K(u^)[oJ'

^(z)

R +K(u) [oj

1. Figures 7-8. Attenuation of Weak Wave Modified by

Previous Passage of Strong Wave of Same Frequency

a. In general the strong wave suffers more attenuation than

the veak one. This is made understandable by examining Eq. (V-l6) of

Section V. IRiere ve see that vhen o) =- v" , the attenuation per unit length

varies as the product of n\^ . Now at low altitudes n ccl/u and \) ecu. However

at higher altitudes n has a weaker dependence on u (Eq. (V-17) of Section V)

and thus the attenuation per unit length is larger.

b. There is very little dependence on power of the attenuation

of the 30 Mc/s wave. This is made evident by the enormous attenuation experienced

by this wave so that very little of the D-region can be perturbed by its initial

strength

.

c. Ihe higher the frequency, the greater the dependence of the

attenuation on frequency. The important consideration here is the level of

attenuation. Since the attenuation varies approximately as l/co , then the

higher frequencies will have greater power at higher altitudes and thus will

be able to perturb the ionosphere.

d. Most of the attenuation has occurred by 8o Van altitude.

kl
lIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIiIIIIIIIIIIL

:: fl 41^:: -85 3 3 W ::

;: to lo 43 -H :

:

:::::::::!:::::::::::::

::;;:::::;:;;:±:;::: ;;

i:^v p a p a u 1;::;!!:::::::::;:::::::;:;::: - *5 I0 5*5^0:: *L 1m U i 11 :

;: 0*^ " S M S M 8 :: CO J CM a d fl j;::;ij::::! :":::::::::::::: : '103 • OhCDh'^ |;::||i:;:::;:::;::::;;::::::: ;

i^ ^ 2 ~ s IS § IS a ;: ::::::::::::::: l:::l!!::::::::::::::::::::::: :

: 3 n

"^ "^ :

:

:: rt 1 :::!!!:::::::::::::::::::::: :

>> 313 • {> + .)

. u« pp V «• ...,,, ........... ........ - -S d :4 1:::!::::::: :::::::::::::::::

-

;; S? :::: :::::::::::::::::t: l:rl!:::::::;::;::::::::::::: :

:: ^

:: «^ ::::l!:::::::::::::::;:::::::: I

;; :s? : i -^ - ::±:±:: t:::::H

:;:l:::::;:::::::;:::::::::::: ±:1-:::::::::::::::::::::::::

''llllllHIIIJlilJimilliilHlJilllllHIIIHIIt-^

.... 1 mJIfflMM::l::::::::::::±::::;:::: /^ + :::::::::::::::!::::: p;;;|; :;;:!: :!-;::::::::::::::::::::: ::::::km::Lj it li ? ::t::::::::::::::::::::

::::::;:.:::::: fflBfBHfl 9

i---i

-[

;;;;;;;;-;|;:: t 1

i::::::::::::::::i:::l! r:::T::::::::::::::;;::: S

::::::::: ::::: :::: 1 !:::: e;;|;;:;:;;;;;::;::;;:;^

...::.:.!!:::::::::::. C

: J :::::::::::

:

:::::::::::::::::::::::::::::::::::::::::;::! :::l! :

:::::::::::::::::::::::::::::::::::::;:: :!:: :::::: I'll — :::::::::::|;;::;:::::

:::;;::: ::::; ;:::;::::! 1 m:: ::::;:::::

:

::::::::::::::::#m

:::::::::::::::::::::::::::::::::!!!::::: + ::::::::

llllllllllllllllllllllllllllllllffllllllM^ miB t^--

TTTMl

;;;;;;;::;;;;;;::;;:;;;; mmmmMmmm^:i |S.- T----

::::!;;; ;;::;::

:^ ^ -- X. .- --- f

E;;;;;;;; |;;+;

yjljU

''-'-t^ .

::±:±j;:|±:::;

.

_j jl |J

.-f

: : :

:

^i:

-lii-lfMl 1 \\\\u

:::::::::::::::::::::::::;;;;;;;:;;::::::;:;: <^ (qp) noT!^

w'.i rr 1 enuaq-ivv illIII! llilWMIMltK

8H

tbO

•H

^ w
BBBBBBBBaaaa

::::::::::::::::: :::::::::::::::::::

!:::::::::::::::: :;:::::::::::::::::::::::::::: :::::::::|:

::::::::::::k:::::::::^::::::::::::::::::::::::::::::::: :::::::::: 1^

:::::::::::: BBBBBBBBaaai aBBBBBBaaaai BBaBBaaaaBBB

::[:|j[[::|| aaaaBBBBB aa

: :: :::::::!

a aa bbbbbbbb

aa BBBBBBBI

aa a aa

a BB aaa a aa

: :: ::: i ::

:::::::::

iii:iii::n::::i i; iiiiiiii III lilllilM iniiililHii

iiiiiiiiiiiii:: i: iiiiiiii iil

iliilllll^ill! li iiiiiiii III

iiiiiil 1

aaa

aaa

'B

a

aaa

aaa

a

a

aaa

aaa a

a

aaa

aaa

a

a

BBB

BBB

B

B

BBB

BBB a

a

BBB

aaa a

a

aaa

aaa a

a

BBB

BBB

B

B

BBB

aaa

a

a

aaa

aaa

a

i

BBB

aaa a

i

:::

::: :I

::: ii: :1

aaa

aaa

a

i

!::

::: :

:

::: ii: 1

!

BaaaBBBBflBB

BBm

BBBBaaaaaa

a

BaBaaaaaa

a

ii-iiiiiii i

BBIBBBBBBB

B

B BB BB BB

a aa aa aa iinilili;illii i; Iiiiiiii ill

lllllllll^llli li IIIIIIII III

aa aa ai :i::ii:ii::ii: „ ii iiiiiiii iil BBBB aaaa bbbb ||||||ft|||||-lllll|U|||| IIIIIIII IIIIII1IIIIIIIII II ItfllllllllltlltllllltlllllitllllllllJIIIIIIIIII

::::• :: ;:: :::::::r:::::H£::::i:::::::::::::i ::::::::::KU:S:K:ia::ii:::S:K::::!::::::::::::::::::::::::::::^

BBBBB a a aaa BBBBB a a aaa :::::::{::::::::*::-- j lltttlt aaaaa a a aaa aaaaa a a aaa :::::::*:::::::::::!::::::::::::: ::::::::::::::::: \:\ ::::::::::::::::::::::::::: ::::::::::::t~~

aaaaa a a aaa aaaaa a a aaa -::;;;;;;;::;:;:':;;;:;;;;;;;;::::;:---;;;:;;;;;- t '.'.'., '.'.'.I'.'.

' -- 1 ^•""

1 I =F J

' ^T^^^^^^ j i i J :-::::-:• a : l-'-ff + + + ^-' : BBaaaaaa bbsbbb ttW^+tt+W HH SSEaSEaSaBB SbbbSSSSbESSSSSE S rSSHSaSSaaS SSBSSBSSSBlaEBBSBlaaBaSSSSSaESESSBSSSSaaEBaBaa^tE"

:;::: ::i : :

:

:>:::::::•:: 1 J r aaa BBBBk.4BBBaB aBaaaaBaaaaaaaaa : BBB BaBaaBBBaaa aaBaaaBaaaBaaaiB a BBBBBaBBBBBB BBBfl aBBBaaBBBa BBBBBaBBBa aBBBBaBBBa aBBBBBBBBB BB^K^^B ^H a BBBBBBBBBBBB BBBB aBBBaaaaaa aaBBBaBBaa aBaaBBBaBBaaBBaaBBBB BB^^^^Bi ^H

::::::::::::

BBB BaBBBBBBaaa aa aaaaa rBBBBaBBB a BBk BBBBBBBBBBB BBaBBBB i ' JBaBBBBa B - IIMllllllMIIIIMIIMIIIMMIMIIIIimMMII -^ -

:::::::::::::

BBaaBBBaaaaaa •a-i*::-

%mm BaaaaBBBBaa BBBa-.BBBBBaaaaBB a aaa BaaaaBBBBBB a-IBBBBBBBaBBBBBB fl 1 5 III 1 lllllll

C 1

' —-

t: .:.:::: A

t::::::l;: !;::::::::::::::::::::j:::::::::::::

aaaaaaBBBei.kBB iBaaaaBB Bazaar

laaaaBBiBi mm .* laaaBBBBBva'^ai

t.::.:::::

:: |)g ^ ^ " :":::::::::::::::

ft 4) S S 2 a :;:::::::::::::::: BBBBBBBB raiBBB BBBBBBBBaaaaaBBBBBBBBBBaaBBBBBBBBBB' .. U M Spa

llllllllllll

:::::::L:: i iiiiiiii::

.i::;:::::::::::::::::::::::::::

:: oJ 1' • "S S "3 S -b :::::::::::::::::

:z <H h ptcupttad o

BBBBBaBBBBBI BaaaBBBBBBBB

aaaaaaBBBaBB

aBaaaaaaaaaB

aBBBBaBBBaBB

BBBBB BBBBB BB BaaaaaaaBBaB

BBBBBBBBBeaa

BBBBB aaaaa Bi BBaBBBBBBaai

BBBBBaBBBBBI aaaiaaaBaBai

BBBBB BBBBB BI aBaaaBBBBBBi

BBBBBBBBBBBI

aBBBBBBBBBBI

a BBBBBaa

a BaaaaaB

a aaaaaaa

a BBBBBBB

B BBBBBBB

iiiiiiiiiiiiililiiiiiliilli

ilillilHIilllilllllilllllllli

" -P r-

:: It

1 ^u mmlm

B BBBBBaa

B BBBBBBB Mlli ::::':::

;;;-:|:;;^

T K ^qP;

00

to •H

VO

k9
2. Flgiires 9-10. Effects of Power Level on Attenuation of Waves

through the D-Reglon

a. The more poverful the vave, the stronger the attenuation.

Ccmment 1(a) above Is appropriate here. Additionally, ve might expect

enhanced electron densities vhen u becomes greater than .1 eV. dls vould

occur at about u/u =» 3.

' o

50
ON

•H

51
^TTTTlllMllllllllllllllllllllllllllllllllll- J :::::::::: ;!":;:::!

:

<

' < i

:: ::; ; P ...H ;;;;;;;;;:;;;::;;:::;::;;:;;;;;;;;;-;;;;- ii ^1 -a -6 *< iA IT 1 m \\

::;

:

:::::: ::: :: A -4- Jt Jt ::::::

i US « IS

:: g -.

: *| S^ 8

rt " O

8 . , .

.

::l::'::: 5 : 61 • . :'|-:::: - H

n a 3 3

: :!:: !:::::::::

,,,,

::l::::::

::l::::::

::l::::::

::l::::::

::

: B o 1 o

i'SS . +

::i::::::: !:::::::::

-

1

1 ::!::::::: ::::::::::

:; :::::: ::!:::::: :;;:;:;;;:;;;;;;;;;;;;:;;:;;;::;;;;:;;;:;;:::

::

3 §9 ::!::::::: ::::::::::

::!::::;:: :::::::::: T "I"'"

: < rn

-+---

: :!:::::::!!:::::::::: 1

l:::±:::

1;:::::::

J ilJ ^

I

ifi w ij!

<

1 ---'I'd

:::::::::::::::::::::::!

:::::!:::::::::::::: ::::::!::::::::::±;: !;:: E in 11 1 m :::::!::::::::::::::: ;:::I;::::::t::::::: ! -::. :::::::::::::::::::::;:::::::::::::::::: Q,. :::!::::::::::::::::

::l:::::::::::::::::l

1 |-

.... —^ - —

1

;;;;;;;;;;;;:;::|:-|-:----^ :

::::; :::::::::: ::::3!:::

- - i- - ' t, —

! :i:::::::::::: :::: i^;

:::::::;::!::::::::: :::::::::!::::::::::: :x:;;:|-::::::::-::-|::-::::::i:^

i;:;;:;:;;;::;::;!;;:;;:

:::!:::::::::::::::: ::: :

:

:

rrrnrnj 1

::::: :::::!:::! ;::;::::!

::::::::;::)!::::!::::::

:::: :::!: I:::: 1:::: :::: ::::;;:;;;;:;:;;;::::

::::::!l:l::::::::::::::

:::::

::::::l:l::::::::::::::: : ::::::ll::::::::::::::::

1

\-

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

; ^m :

!

E^^^^^^^^:^^^;;=;;p;":;;;i:;;;:;;;;:;;;;; ;

::::::::::: ::::;:::::::::::::::: ;;::i"::::";;:::::::: S

:

:

:::::::::::::::::±::::: ::::::::::::: 3: :::::

:

':':l':':':':i --

#1 II 'l ||-L|.||[|

:

;;:::;|-||:

.... ^t ;:::::::::::::::::::::::

:

:::::::::. ::;:: :::::::::

:::: z-r::::r ±:-i m i

::;:::±:::

1 1 1 1 1 1 1 1) 1 1 1 1 1 1 1 11 ...i < 1 qp 1 T^f""*"""""

,^

::::::: ::::::::::::;

1441 ;

w•H

I

52
3. Figures

^ - 12. Attenuation of Weak Wave through D-Reglon

Modified "by Previous Passage of Powerful Wave for Several

Power Levels

a. Little dependence on power level Is noticed for the powerful

wave of 30 Mc/s and 4oV 2 Mc/s. This can be attributed to the high level of

attenuation suffered "by the powerful waves at these frequencies.

b. At 80 and 320 Mc/s u(z^)/uq(z-|_) = ^.00 produces the most

reduction of attenuation. u(z, )/u (z.) = 6.00 Is not effective as we shall

see later because of n /n => 1 at certain altitudes.

00'

The curve labelled "optImuirf' In Fig. 6, Is obtained by

maintaining the attachment coefficient, K, at Its maximum value at all

altitudes

.

53
•H

54
to

•H

55
k. Figure 13 . Attemiation of Weak Wave throu^ D-Reglon

Modified by Previous Passage of Wave8> for Several Frequencies

but Same Effective Power Level . ""

a. Tbe greater the frequency^ the greater the reduction of

attenxiation. This can be attributed solely to the fact that the absorption

of the powerful wave decreases as l/o) and thus the higher frequencies can

penetrate to higher altitudes with less of their power attenuated.

b. The maximum reduction of attenuation occurs iidien

u(zi)/u (z^) = 'i-.OO. We can see fz^om the slopes of the curves that most of

the attenuation occurs between 6o and 70 km. Thus the power level

u(z. )/u (z ) m k,QO which maximized the attachment coefficient at 6o km

should cause the greatest redaction of attenuation.

c. The curve labelled "optimum" In Fig. 13 is obtained by

keeping K, the attachment coefficient^ at its maximum value at all altitudes.

56
I I I I I I I M I I I M I I I I I I I I I I I I M I I I I I I M I I I I I

I

I

. J I I

I . I I I M I 1

57
5. Figure ll». Perturbatlcaa of Avgrage Electron Energy In

D-Reglon due to Presence of Waves for Several frequencies but

Same Effective Pover Level

a. We observe a peak in u(z)/u (z) at about % km altitude

for three frequencies. These are e^qtlalned as follovs: u(z)/u (z) obeys

the equation

JL. = i +

u.

8 X lO^^E^ fel

8 1

At If 5 Inn altitude v>^ = 1.22 x 10 sec but when u/u = 2.52 o o

Q -1 then V = "»; X Vu = 300 x 10 sec . o ' oFor f = 30 Mc/s, (i)/t> = . 94, g^/oi = .25, g^p = 1.3 and

2 2

g^ » 1.2 (o) •<- g.) . (See Section IV, Figs. 2 and 3.) Nov as the altitude

Increases V decreases (approximately exponentially), the denominator on the

rl£^t decreases, g /\7 approaches 5/3 end g approaches zero. Thus u/u

vlll increase until the effects of the collision frequency In the denominator

Is no longer felt. At 51 km the collision frequency is down by about a factor

2 of 2. Beyond 51 km the effects of the inverse square dependence of E on

altitude and, more importantly, absorption become apparent (see Figs. 1, 2).

When f = 80 Mc/s cd/ v^>^ 5/3, g. / uo= .15, g / u = 1. 42 and

1 r 2 2 gL =1.83 (g. + 0) ). Thus u/u still depends on V but not as much as for

f = 30 Mc/s. This is shown by the lower peak for f = 80 Mc/s.

2 2 When f = 320 Mc/s, g -c « (u) + g.) and thus there is no dependence

r 1 on V and no maximum. However, when u/u =6.00 then iJ Is sufficiently large

58
80 as to make Iteelf fielt as a small maximum at kQ Im. Cie secondary

maximum at 78 km occurs because of Iccal variations of u . o

b. Hie greater u(z^)/u (z ) (i.e.* transmitter power level),

the hl^er the altitude at which the maximum occurs. Ibis is evident fraa.

V depending linearly on u/u and the previous discussion.

c. The hi£^er the frequency, the lower the altitude at >Aich

the maximum occurs, ^is is a very weak dependence.

d. Th» dependence of the magnitude of the maximxm fj^equency

decreases as u(2l. )/u (z. ) at k^ km increases. This is also because of v'

varyl'^S *' V%*

e. u(z)/u (z) is greater than one at hig^ altitudes only for

the hig^ frequencies. Ibis results from the absorption dependence on l/(u .

f

.

Hi order to obtain the most efficient reduction of electron

density u must be kept at -^.09 eV or u(z)/u (z) ^^ 3 between the altitudes

of 60 and 70 km. Of all the examples chosea, f « 320Mc/s and u/uq = ^.00

appears to accomplish this best •

59
60
6. Figure 1^. Perturbation of Average ELectron Energy In

D-Reglon due to Presence of a Wav^ for Several Pover Levels

a. At higher altitudes u(z)/u (z) Is a nonlinear function of

u(z )/u (z ) at k^ km. u(z)/u (z) for f = 320Mc/s between u(z-^)/uq(z^} =2.52

and u(z )/u (z ) = ij-.OO Is an exception.

61
III 1 ^ s Ml 1

,j5| " * * ::;:;::;;;;:;::;:::;;;;:;;;;:;; Q «»1h i/N O O :::::::::::::::::::::::::::::::

:: 49 do • • • ::::::::::::::::::::::::::::::: S, ft N jt vo i 5 b «j 1" " n mill HIIIIIIII 1 lllllliTHn

::::::::::::::::::::::::;::::::::::;:::::: J ::::::::::::::: ::

^ S" -*T J i-"- £' •±--"*"" 5^' -Jt* X

J.! j J.

|:::::::::::::::::::::::::!;::::!:::::r ;::::::: :::::::::::

xi — i'--Ji--

ill

B « ttSs 3 3 3 3 3 3 ttitfl fitt HUfHUffl ::

i' Is .5 . -c^ + :;;;:;::ii=-l:;--;^;::l-- ^^u|:^il 1 i||[f[{||j|i

1

1 i^mi :::

; ;5^§ • ^ + ::::::::::±::::|::|::::::::: H iH - - - - - - - - - ::::::::: ::::::::::|::i::::::;;::::|:::::::::::::::::: + :| ::

:

\ ^ >» W) ::::::::::S^T-"I-""":: ::

;

R r f^ C 1

1 1 yimwlti X -- 1 j-T.--i.-T --

::

:

; P8^ :::::::::::::::::::;l:||::: :*:*:::::::: xp::::;*:;::: ::^ :::::::::: ::::::::::£:::: H <^ III III:::::::::|:jS::J::t|: ::::::;::::::::::::::::::::::: T

:

S ^£1 Wiiiii:::::::::::|::g:;:|:::--;:::::::!::::::::::|:::::|::::

'. '.

'.

;;;::;:::|::;il|:i;.;;;:::;;:::;;;;;|;^^^^ \ ;

;:::::::::::::::::::::::::::::::::::::::::::::r::::::::"::":::-"|!!":iW#MMWM^^t^^^ ffl

; ::::::;:;:::;:;;:;;;;:::;;::;:;;::;:;l;;::;l|:::;;:-::::||:|::; trtftt^ttt^tJ##t4#ti#ttti Hill f\ -^S "i -ttf^t "± '"'T ": • ; :i:::;:|:|i^:^gl:p|;|::::::::::::::::::::: «

< =-;|:::||:;4|:|:i::^-|:::;:::ii::::::::;::'^3

3

: #wffWiMrn#^n'W M I'W-^S ::i^:::::::^ii:±-i#:ilii:::T:4:-"^S^:::::::J> :

1

1

1

1 l^^^^^l

;;-|;:i!;;;;;;i;;;|i|;plj:;|;:;;:;;;;;;;:;;;'J : :::::-::;:::::i|.ibi^. -L:^:tfj:-y-x:::::|:::: ...

.

::-::i-:::--;:-g-:-:i'g::j-^S::j:; :::::::::: ::::::: ZZ'. ±:::::::;;:::T-;-tiTtx::::±3 ±E± x ':':'

::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::l 4 il#ltrmj^fflll4^^l?ftwntnt ti n1 ft+tttt ::

:::::::-:--:-::::::-::-:::::::;;::-;;::::::::::::- M 1 '•

1 1 fr 1 fl -\ hrflttj H M ' ±---il'-^±± ---- :::::::::: :::

.,, r-. ... :;::::::::;:::::::::::::::::::::::;:::::::::::::::::::::::::::::::::!::::::: -::: I": ::::::!: ::::::":: :!:i :::::::::::::::::::: ::::::: ::: jfjjM- tt 1 ::;

:::::::::::;::;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :|::|q-:S!:::::TT:-:::::-r:-:^:-- ::::::::::::::: ::::::: ::: + -r- + -- L .___. ^ h^i -M ^ :!: itlm it"""T T^^"^—

r

tT" ' + T

::

;:::;£SlSS;;::::;:::::::± :::::::::::. ;;:::::;i;:,;*- -T--Trp-n

:::

:::

^-»F ITN

00

62
7. Figure 16 . Perturbation of Electron Density In D-Region

due to Electromagnetic Wave for Several Frequencies but Same

Effective Power Level

a. The local maxima In n/n at 51 km coincides with and are

00

due to the maxima In u(z)/u (z)

.

b

.

The magnitude of the minimum value of n /n /«^ . 57 is due to

00

the linear extrapolations of K(u ) from Biondl's data. Had we used

K(u ) = 1.5 X 10' cm then n /n minimum would have been .33. o 00

c. The D-region is significantly perturbed only by the 320 Mc/s

wave. All the others are excessively attenuated.

d. When u/u = 6.00 at 45 Ion the local maxlmiam for frequencies

less than 320 Mcph&s values greater than 1. Ihls occurs because u/u is so

high at this place that K(u) -^ K(u )

.

e. The curve labelled "optimum" is obtained by

maintaining the attaclment coefficient at maximum value for all z. We see that

the 320 Mc/s wave at u/u = i<-.00 approaches this curve remarkable well.

63
1

::::::::::::: -S m J

:::"q: ::::::::::!!!!!!!::: ::::::::::::::: ::T:::::g::

---++-T 1- i-±

Ill III 1 iiiiii 1 II iiiiiiiiii 1 inji ijii

::::

:-- ;;;;

:::!::::::::: ' ^^ o> N '-> II ::: t:::::::: »-' n o

O vn o O 2 :::::::::::::: ro iA 00 CV3 -g, 1 m g* ::::::::: :::::::::*:::

O ::::::::: ::::::::::::::

:::

§

|l 1

+ - ..q:-- + _. ::::::::::::::

::::

::: ::!::::::::::::::::::::: i::::::: rt IT

:: Mr: :::!: :::::::::: :::::::::: :::: :::!:!::::::!: + ::::::::::::::::::::

:::!::!r::::::::::!!::::::::::::::::

:::!::::::::::::::::::; ::::::;;;::±

:::!::::::!:::::::i::: :::::l:::::±

::::::::::::; :::::::::: ::i: ::::: :!r -- I-- ±-

:;;::::;::::::::;:::::: + :::;;::::::|::: :;l: :^::::::: Ml iiiilii :+> : 5!

:;::!::::;;::::;:;:::::::::::::!:::: :::::::::::::::::::::;:!:::::I::::::::::±:::::::::I:: ::5:::::i::::

::::!:::::;::;::::2: :::::::::: :!:::::::: :::: J:: 1 :;:!;::;:::::;: ±: J 5 :::::::::::::::::::::::::::l !:::::::*: :::!:::::: !5:i I::::::*;:::: MlX

1 - - H-1 : flii^flMl !-. - ::::::::±:::;:: ::::;;:::: !::::! !!::::::: ::r:

:::::::T:::::-:::<f:l::r:::i:::":::::: ::::: ::::::::±::l!::::::::::::l:::::::::::::: ::::: :;;:;:::l!::::::::::::;l::::;::±;::::::: ;:;;: ::;:::I!::::: + ::::::I::::::::::::::::E ::::: :;::::!;;;::::::::||;;:;:;;;;;;;::::;:;;;;

3g;;;;;|;|^;;;|:

:::::;:::::i::;:::;;;::i±::::::::

:|:::::!l::::::a:.:::i:::::::::;;:±::r::;:

:!::;:; :.:'ii;:::::;:::::::i:::::;:l::

ttl 1 Wn Mil n^tfl 1 M [fllr

ff'^

at

f-|---

f -T---

^:l _.i -

s

kI r

^

^..± T i--;.± :::::::: Mili^M::::::

::-

_,.._

\^l

m

:i-:::;:x::it:t-:i-:-LlSL:i:

.-|;|i|:;::-:;:|;::p;:;;;|;:;;; ----I i

'::\ :::

::: :;;;

-- -

VOH

•H

&^
8. Figure 17. Perturbation of Electron Density In D-Region

due to Wave, for Several Pover Levels .

a. The greater the power level, the deeper the region ^Ich can

he affected hy the powerful wave.

h. The optimum power level, as is made clear for f = 320 Mc/s,

Is u/u = it.OO.

' o

c. See comment in (T-e) ahove.

65
«1 Sv 5

f liliiiiliUijIill^^

::::::::: ::;;:;:::;

;:::;::::::::::;:::::::::::::::::::::;:: 1

wavi

icall; densi'

:: <S 1h O e B 6

:.t:::::i.:::i::: _ u » p ' 3 M S

- 5 4» i» lA UN IfN

: :--::-:::•: I : --

T rH U -P tf 4> q -P « _« * •g IS

= « 1 i3 ^ K S 8

^:::::::::::::

::::::::::::::::::: ;:::::::::::::::::::

::::: ;:::::::i :::::::::::::::::::: [;;|;;:::::

= 0) P< U 11 II " 3 j :::::::::::::::::::: :::::::::::::::::::

:: 1 -S s -i . o 1 O 1 O Pt : :::::::::: ::::: ::::: :::::::::::::::::::

:::::::::::::::::::: ::::::::::::::::::: :: i5 ^ 5 ^ '^ =* ^'P ^P ° f

F P O W tJ

=;;;;;;; ;;:;

::::: ::::: ::::: ::::: ::::::::::::::::::: Ji ^ g ^

V J ^ ^ -' - E :»::::::::::::::::: » >» Q 3 t? + w E

- ^-v fl) tl J)

:::;:i:::::::::::::::::: :!:: ::::!::::: It Lyii ffffl

:; a^F^ 5 fl®

::::::::::::::::::::::::::::: ::::::::! :::::::!;:::::::::: ::::!::::::t:I!::: Tn 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i4ffl BH JD 1 1 1 ::::::::;::;:::::::::::::::::

::-;;i::::-::t|' :::::±::l:::::::;:::::::::::::::::::

::::::::::::::::::::::::::::: ;-;;;; |:

:::::::::::::l::::i:i::::;:f±::::::: ::::::::;::: !:::;!:::::

::::::q:::::::::::! :::::::::::::::::::::::::i::::::i:|

::::::::!::::::::::::::::::l::::i:::

r- ..-

ii :;;;;:;:;;;;;;;;;;;;:;;:;:

....,...,| =--:;-i;:!-r;;;;;i;;;;;;: w\::::::: H :: ::::::::::::::::::::::::::::::::::::: ' [

«

r --! ±::::::

: <

::::::::(: :::::::::: ::::: ::::: ::::::! :±!;:::::::::::::;:::::::::::::l::: ::::::::::::^::::::::::::::l:::::::

: I :::::: :!j ::::::::::::::::

:

::: :::::::!! ±.

:;:;;;;:;;;;;: ml t:::::::::!!:::t:: :;;::::;;;;::::;:;;-;;!;;:;:;:;;:

::: ::::::!;:::::::::::: ::::::

::: :::;: ::±:::::: Hi 1 1 1 1 1 1 1 1 1 1 11 1 1 1 Itl 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

:;;;i;;;!i;;;;:i:;;:;;; ;;;;;;

::::::::::::::::::::;;;;::::;:;;;;;:;;;

:l:::±:::::::::::: :::::::::::::! ::::::::::::::::::::::

:::::::::::::l: :::::::::::::::::::::

:::::::::::l::::::::::::::::::;:::::: ::::::::::l::::::;:::::::::::::::::±

1 ---

I'l-^^T-^ -::::i±:!:::::::::::::;:::::::::::: 4- t"i

::i::;:;||;|;;;;;;;;;;:-:;;| :: + :#:::::::::::::#:::::{::::: + :: J : i: ::::::::: :t ::::::::::::

:

^^^H 1 1 [tl J 1 M

±_-

4T=FF"-faa:- "§"" ± ft"' WW|^;|i;;;;:;;;|;S;;±;;::;|]

::;::;:::: :::r ::::_ ^S::;:: i:::::

lljjj||[|||y|l-^

1 H 1 1 11 1 1

1 'ffiifM i

1

1 1 n^

-ff;:5;;;:;±;;;;;±;::::::

:i:;;;:: ;:;; r rir:::::

;:::t:-::-:- -++ 1+*-- -+H "vfT mmllii-iii

;::::::-|- r-^^n5T^^

i:;:;;;;j3m

w<— ^ mH:

;

" ; — + i " :

" : :

:

:: J

:;:::.:::

:

bO

66
C. Qualitative Conclusions Obtained frcan Examination of Figures T - IT .

1. A veak wave propagates in a significantly different fashion than

a strong wave.

2. Attenuations vary approximately as l/a> . Thus, nonlinear effects

Just now demonstrated are more spectacular for lower frequencies.

3. Attenuations vary as n(z), thus, the nonllneax effects are more

spectacular for large n(z)

.

he Region of maximum attenuation for a veak wave passing through a

perturbed D-layer should he at an altitude >rtiere K [o \ /r j^ 1 in order

that we may facilitate propagation through this region.

5. The perturbing wave should be at a high enough frequency so that

it Itself suffers little attenuation,

6. The pertiirhing wave should he at high enough power level so that

K (attachment coefficient) is maximized at the region of maximum attenuation.

7. The lower the electron density the higher a strong attenuating

wave can go and still perturb the electron density. The res^lltlng perturbations

depend In minor detail on the magnitude of the electron density, but more on

the distribution.

8. There is an optimvmi power level for the perturbing wave.

9. Ihhomogenelties in electron density may be induced, or already

existing Inhomogeneities may be altered by the presence of a powerful wave.

10. The electron density cannot be reduced by this method

above an altitude of ^—^ 8o km

.

67
D. Analytical Approximation to the Absorption of the Wanted Wave

We have seen that the best that can possibly be done to facilitate

the propagation of a vanted vave is to maintain the attachment coefficient at

its maximum value of ^.0 x 10* cm . By using this strategy one can reduce

the attenuation of a 30 Mc/s wave from 84 db to 52 dt. We gave cautioned,

however, that such estimates of reduction may he conservative since we have

used a rather large value for K(u ) based on a linear extrapolation of Biondi's

data obtained at a higher temperature than prevails in the D-region.

In this Section an cmalytical approximation to the reduction of

attenuation is developed to show how it depends on the asstimed value of K(u )

(or equivalently, K(u )/R).

IQae altered attenuation of the wanted wave is given by the following

expression:

100 km 2

Attenuation = — / j

2—

L

[(<»i + gj)' + g.r

,

, ^kA^zY

"^

K(u)[(

LI + ^

dz. (VI-1)

Now let us assume that u is constant over our region of integration and k(u) ° r ]

r 1 -(z - z^)

is at its maximum velLuc. Furthermore, g^ -«: cd and [O^J = LOgJi « r •

We observe In Table VI-I that nv> is fairly constant in the region of its

maxlxmmi values, from ^ km to TU km. Assuming the constancy of nV over

this range we can write (VI-l) as

68
Attenuation = 2—_i

ecu /.

"2^

1 + a exp

1 + "b exp

>2(z - z^)/l]

-2(z - z^)/Lj

dz (VI-2)

where a = K(u^) [OgJ^ / R , b = K(u) [OgJ^ / R, z^ = 5*^ km, z^ = 7^ km and L

the scale height at 6k km is 7.0 km. Then by simple transformations and integration we arrive at

2 , Attenuation

CO) V> 1 + b exp

1 H- b

-2(z2-z^)/L-

+ 2

)

(VI-3)

The linear extrapolation of Biondi's data yields a value of a = 58*^ and

at K(u) , b = 103 at 5'*- km. By putting in the value of nu of

11 -3 -1

20 X 10 cm sec , one obtains a value of attenuation equed to 51'5 <3b,

remarkably (and probably fortuitously) close to the computed optimum of 52.31 db.

Now -vrtien we take Nicolet's estimate for K(u ) of I.5 x lo" cm /sec then

a = 12.3 at '^ km. The attenuation now becomes 30.0 db.

Thus, we see that our predictions of attenuation depend very critically

upon the values of the attachment coefficient. Taking the optimistic point

of view, we can reduce the attenuation of a 30 Mc/s ware from 84 db to 30 db,

a 5^ db decrease or an increase of power by 2.5 x lO''.'

69
E. Pover Requirements

In this Section, ve have shown that a poverful vave can alter the

electron density In the D-region and modify the propagation of a vanted vave,

sometimes decreasing and sometimes Increasing Its attenviatlon. The demonstrations were hased on a Gedanken experiment \riaere It vas assijmed that a

wave of sufficient power existed already at the altitude of ii- 5 tan. We now

Investigate the power requirements of a ground-based transmitter to produce

such effects vertically at 45 tan altitude.

We neglect magnetic field effects. Then Eq. (IV- 24) of Section IV

becomes

:

2 2

H-. = 1 +

g^ ^r^o 1

. 7.9 X 10^9 gr^o . ^^x u 2mGVu f/ s2 21 ~ Gvu / x2 2 ^

' o ^" o [(oi + g^)^ *" 8r J ° (a> +

gi) + 8^.

with u In eV and E In esu. o

Let us ass\jme G = 2 x lo' and V = ^^ (z) — ; also at U5 tan o u

o V = 1.22 X 10 sec ^; u = .0358 ev. Now taking f = 32a Mc^ then a)/v'*»-l

and g, <: *: CD, g = 5/3> co = 4.1 x 10 sec" . Ohen:

1 r ^ = 1+4.5 xlO^EV"^ . (VI-5) u o

o

When u/u = 4, then E =3 esu/4.5 x 10"^ = 6.7 x 10 esu.

70
c 2

Now jj^ E^ Is the power flux at this altitude. The power output of

an isotropic radiator to yield such a flux is given "by

^

E x kitr^ = cE ^r^.

•4-Jt O O

Ohus, to obtain u/u^ = U at U5 km (r = 4.5 x 10 cm) and f = 320Mo/Bwe need an

isotropic radiator of power output 3 x 10

10 x 6.7 x lo'6 x (1^.5 x 10

6^7 ) x lo" 'watts

= l|-.l X lO'' megawatts. A 300 ft parabolic antenna would give a gain of k8 3b

with a beam width of about .75 degrees [ SRI, n. d. |. This would cut the power requirement to 6.5 megawatts. The region -vdiich would be affected at k^ km would be

rather small, however, only about ^ km in diameter. The power requirement for a 4

100 Mc/s isotropic transmitter would be reduced to about 7 x 10 megawatts.

A kOO ft parabolic antenna would provide a gain of kO db with a beam width of

o

'^ 1 . 5 . Ttxe power requirement would be reduced to 7 megawatts and the region

which can be pert\irbed at U 5 km altitude would have the extent of about 1 km.

Oius, in order to perform experiments of the type described In this

section, we need high gain CW (pulse length 1/2 second) transmitters of

frequency 100 - 300 Mc/s and power output of about 10 megawatts.

Transmitters of nearly such power levels, directivity and frequency

are already in existence or being constructed. For example: The ionospheric

radar probe being built near Arecibo, Puerto Rico [ GSordon and LaLonde, I961]

operates at 1^30Mc/s,2.5 megawatts peaJc, 150 kW average, 100 KW CW, 60 db gain,

maximum pulse length 10 sec. The transmitter at Lincoln Laboratories field site

,0

at El Campo, Texas, operates at 38 megacycles, 500 w; CW with a i beam width

in one plane and 8 In perpendicular plane. The Lima Radar Observatory in Peru is

designed to operate at 49.9 Mc/s -with 6 megawatts peak, pulse duration of

5 X 10" sec, gain of 39.7 db, TDeam width of 1.07 by 21.4 . Now the

* The D region is within the near region of this transmitter so that even when

operated CW it can seriously modify electron energies at 60 to 80 km altitude. ** Private communication from Martin Balser. *** Private communication from Lima, Bsru.

71
power estimates are "based upon Eq. (VI-U) where it is assumed that GV ~' u/u . According to the work of Altshuler 1963 it would appear that G))^^ (u/u ) -1/2

If this is actually the case, then our power estimates may be overly pessimistic.

F. Communication Facilitation during Blackout Conditions

By a series of Gedanken experiments we have shown that the

propagation of a wanted wave through a very heavily perturbed Ionosphere

may be facilitated by modifying the D-region through which the wave passes.

The modification is accomplished by illuminating the region with a high

frequency powerful wave. In our Gedanken experiment the wanted wave was

of 30 Mc;^ and suffered an attenuation of 85 db when propagated vertically

to 100 km altitude. If the D-reglon Is illuminated by an intense 320 Mc/s

wave of strength such that at U 5 km the energy of the electrons Is Increased

by a factor of k then It Is modified so that the 30Mc/fe wave will now have

an attenuation of 57 db.

The modification is accomplished by heating the electrons, Increasing their attachment rate and thus decreasing their density. In our

computations we ha-ve used very conservative estimates for the attachment

coefficient in the D-reglon. Had we used the currently estimated value then

we would have obtained a much lower modified attenuation of the 30 Mc/s, wave

more like 30 db. Thus the efficacy of the technique described depends

critically on exact knowledge of the values of the attachment coefficient.

The computation of transmitter requirements suggests a 6.5 megawatt

transmitter with a gain of U8 db. Reducing the frequency would reduce the

72
power requirements by approximately one-half. If the frequency of the

powerful wave is reduced too much it may suffer excessive attenuation before

it reaches that region of the ionosphere it is to affect. The power

requirements are based on a crude theory of inelastic collision processes

affecting the average energy of electrons. Based upon recent work by

Altshuler 1963 these power requirements may prove too large.

The propagation of the wanted wave is accomplished only in a very

short-time interval ( ^^ l/k sec) after the powerful transmitter is shut

off. Thus a system that would use the above experiment to facilitate

communication between two ground points would be able to communicate only

within short {l/h sec) but closely spaced (l/lO sec) intervals. Furthermore

if both ground stations were suffering from blackout due to a perturbed

ionosphere^ then both stations might have to have powerful transmitters.

Under these conditions if the two stations wanted to cooamunicate via the

E- or F-region^ both powerful transmitters would be pointed along the

propagation path of the wanted wave and both would have to be turned on

and off synchronously. Because of the very high gains of the powerful

transmitters, very narrow ducts would be created in the D-region for the

propagation of the wanted wave. Thus very accvurate positioning and

orientation of all tra. ^xtters is required.

Our ability to modify the propagation of a wanted wave depends on the

distribution in height of the electrons. If, as in our example, the

region of maximum attenuation is below 70 km^then we can accomplish a

73
reduction of attenuation (in db) "by perhaps a factor of 2 1/2. If the

region of maximum attenuation occiors above TO km, then ve can exercise

very little control over the propagation.

When (0 =- 1> the region of maximum attenuation occurs at the altitude

where the product nV meoclmizes. As the frequency is lovered the position

of msLxiraum attenuation for the same distribution rises slightly, lima for

lower frequencies the procedure of reducing electron density is at some

disadvantage as ccmpared with higher frequencies. However since overall

attenuation (in db) varies approximately as l/a> and since the fractional

reduction of attenuation is just slightly dependent on co, then the propagation of low frequency waves is more greatly facilitated than high

frequency waves.

Because of the cost of the required powerful transmitters we do not

at this moment recommend their employment as a practical means of reestablishing communication during blackout conditions. It has been suggested

by Molmud I963 that they can "be employed, however, as diagnostic tools to

determine the properties of the lower ionosphere, even during conditions

of blackout Molmud, 1962b

* VHien this paper was written it was not realized that the D region was within

the near zone of the Arecibo transmitter.

1^
VII. SUMMARY

The object of this study is to determine to what extent the propagation

chaiseteristics of upper atmospheric plasmas can be ameliorated by powerful

groTind-based transmitters. The investigation centers on the role of electron

energy in determining steady state electron density.

The important processes controlling the electron density in the D-region

during daytime are ionization due to U.V. , X-rays and protons from the sun,

attachment of electrons to Op and subsequent photodetachment from the resulting 0~ and also recombination hetween electrons and 0~ and positive

ions.

The attachment coefficient for electrons and 0- Is monotonlc Increasing

from .ok BY to y^.l ev vhere It peaJos. Cie ratio of the values of the

coefficient at these points Is —2. Thus, in an otherwise steady state

situation and in a region ifhere attachment rates dominate detachment rates,

the Increase of the electron energy by a factor of about 2 should decrease

the electron density by a similar factor.

In a situation vhere the electron energy is suddenly changed to a new

and constant value, the electron density relaxes to a new value In a

characteristic time depending Inversely on the svun of the attachment,

detachment euid recombination rates. The relaxation time depends on the

75
energy of the electrons because the attachment coefficient varies with

electron energy. Tiie positive Ion density changes also but at a much

slower rate.

The electrons In the D-reglon can be altered In average energy ^en

a high-powered electromagnetic wave Is present. This field-Induced

modification of local conductivity necessitates the solution of the

corresponding nonlinear propagation problem.

The conductivity of the medium in which the wave Is propagating

depends on the electron density and collision frequency and these , In

steady state, depend on the field strength. The conductivity Is obtained from an expression for the average drift velocity of electrons.

The drift velocity and average energy of the electrons obey Intimatelycoupled nonlinear differential equations. Approximate solutions of

these equations show two different relaxation times: about 10 sec

for drift velocity and about lO" sec for average energy (in the D-region)

.

In contrast the electron density relaxation time is about lO" sec.

The propagation of an intense electromagnetic wave through the

D-region will heat the electrons and therefore alter both the electron

density and collision frequency. The optimum reduction in electron densityoccurs at about 0.1 eV for the electron mean kinetic energy. During

blackout conditions a megawatt transmitter with kO db gain at about 300Mc/s

can decrease the attenuation of a wanted wave by a factor of two to

three in db.

76
APPEMDIX I

PERTURBATIONS DUE TO MASS TRANSPORT AND DIFFUSION

We are Illuminating a small region of the D-layer hy a fairly directive

beam of powerful radio waves. The gases in this region may have velocities

in the nei^boiiiood of 100 meters/see [Handbook of Geophys,, 1957»] The electrons

because of the inhomogeneltles In density and energy created by the powerful beam

will be subjected to perturbations arising out of these Inhcmogeneltles. In our

cOTiputatlons we have not Included such effects on the piresumptlon they are negligible

especially since the perturbing beam is on only for less than one second. We now

contpute the effects due to mass transport and diffusion.

Let U8 rewrite the moment equation including the term allowing for

spatial Inhcmogenelty:

^(n5) -V^ . (nvx) -n V, — E + 01 X V m %

(1)

= r (X' - X) F(V)f(v) |v - "Vl ff ( 0, 1^ - ^1 ) d^d^vd^V .

Then the moments: density, velocity and energy become (assuming air)

^ n +V'(nv) =

^(nv).V-n^%jj^ Vn

^1 2 V ^mv w • V^ m

(2)

(3) = f (E -^^xl^)

11
(i^)

We first observe that mw • V, for a Maxwellian distribution is just

-, ,-. Also, V • rivv' = — 3d ' m

—^ V . Also, V • nvv = _* * vbere p is the pressure, l/3 nmv . Hovever,

2

at low altitudes n is perturbed by the powerful wave such that n CC l/v

(i.e.^ K CO l/u for low values of u) . Thus, the electron pressure is not

affected by the powerful wave and the second term of Eq. (3) can be ignored.

Now temporarily ignoring the term 9n/3t we can consider (3) an equation

in v - V, and solve it for this relative velocity. Here V. might be a wind

velocity. We will get solutions similar to (lV-15) and (lV-l6) of Section IV

with the added complication of scxne additional terms due to the interaction

of V and m . These terms are: (to be added to right side of Eq. (lV-l6)

Section IV)

-V 01 (Osin (At - m cos oi t) - V % (^ cos oi t + oi sin oi t)

2 ~2 '

(to be added to right side of Eq. (lV-15) Section IV)

Oi V (i) cos 01 1 + 01 sin oi t) - V {V sin oi t - oi cos oi t) oi

djr.

2 2

(Eq. (lV-17) is unchanged.)

78
We see that v - V , might suffer periodic perturtations of magnitude V. d a

if 03^ ft^ U . "We have not included all the perturbations on the electrons

however. Ttie positive massive ions ohey essentially the same differential

2 2 equations as the electrons. However, for them ax -= < v

^

"U^ D-region.

Thus their average drift velocity is unimpeded hy the magnetic field of the

earth. If the electrons underwent any appreciable motion differing from

those of the positive ions then charge separation would ensue which vould

cause large field greidients. IQiese fields thus cause the electrons to move

with the same detailed motion as the positive ions. For this same reason

the diffusion of electrons out of or into the perturbed region will be

seriously modified by the presence of the positive ions.

The diffusion coefficient required to describe the diffusion of a

mixture of electrons and positive ions is the ambipolar diffusion coefficient

[ Allis, 1956] which for our purposes, can be represented as

a

" M V^ 3pQ

'

Now since Q is proportional to v for electrons in air, v/q is insensitive

to temperature. However \) varies as the temperatxire of electrons. Thus,

in Eq. (2), nv = U D n but again, since D varies as the temperatuare \rtiile

a a n varies inversely as the temperature at low altitudes, then, the diffusion

current induced by altering the electron density (due to heating) is negligible.

The term nuv in Eq. (k) may be treated as s^ nuv and then

V » nuv = uV • (nv) + nv . V ^' ^« first term in negligible, from

79
previous considerations. The second term will give time averaged

contributions only from the steady drift motion of the electrons (carried

by the winds, usually horizontally) and from the conponent of Vu in the

same direction. Now Vu in the wind direction will arise mainly from the

radiation pattern of the ground-based transmitter and at D-layer altitudes

will have magnitudes of the order of electron volts per kilometer. Accordingly the contribution of V • nuv'will be negligible in (h)

,

w V V . V^ M -- The term :? = may be approximated blb Ci:,-^ y • V, , ^ich \rtien

V , 2 time -averaged yields i^V, , or the average energy associated with the

drift of the neutral particles. This energy amovints to .0015 eV for a

100 M/sec wind, a much lower energy than .033 ^ the mean energy u of the

gas particles themselves.

Thus we see that mass ti^nsport and diffusion have little wei^t on our predictions of the effects to be encountered by a momentary (i.e.j~'l second)

irradiation of the D-region by a powerful beam of radio waves. The decay of

the coliimn of reduced ionization produced by the beam, after the beam is shut

off, will be controlled more by photodetachment than by diffusion. This is

because of the very small values of D and Vn (i.e., 9n/9t = DU^n and

at 60 km altitude D?:»2 x 10 cgs, V ^n -- 10^10"^° cgsj therefore

9n/9t^=^10 cgs, whereas due to attachment and detachment 9 n/9 t '>> 10 cgs).

We are of course assuming that the region illuminated is sufficiently

large so that the electrons take a significantly longer time than their

density relaxation time to be carried through this region by any wind at the

altitude in question.

We have not Included the Inelastic collision terms In (14-) . Its dependence

on mass transport may presumably be treated by the techniques described by

Altshuler I963I .

80
APPENDIX II

AMBIEMT ENERGY OF ELECTRONS IN D REGION

We have ass\imed In our paper that the ambient electron energy in the

D region Is the seme as the background gas. Sears 1 19^3 1 and Kaz*zas and

Latter [ 19^3 J have suggested that electrons -when produced are fairly

energetic (3 ev tram photodetachment, 10 e\r from Ionization) and therefoi«

that electrons found In the atmosphere may have an average energy significantly different from that of the background gases.

Let us examine the effects these energetic electrons may have on the

electron average energy.

We again work vlth the transport equation and its moments. The transport

equation now appears as follows:

/ at

i^ + Rn_<f>(v) + qf(v) - >J2(v)f - V2(v)f (1)

where the ncanenclature is the same as in Section IV, <P (v) is the distribution

function of the Just photodetached electrons, q is the ionization strength vlth

^(v) the distribution function of the collislonally (by ^ particles say)

produced electixins, 15_ is the 3-body attachment rate and V„ is the two-body

dissociation rate.

We now multiply (l) by u, the kinetic energy of the electron and integirate

over all of velocity space. Assimilng steady state we obtain (as in IV-23)

nGv(u-u ) = Rn_u + qu. - uv>-n - uvLn , (2)

where nGV(u-u ) = u -^ and Gv is the energy relaxation rate, empirically

deteimined. u, is the average energy of the nascent photodetached electrons, 3 ev

d

81
and u. is the average energy of electrons resulting from the ionization

process --^lO ev.

It is evident, now (ftom Eq.(2) ) that — Rn u- u.

IL. ^ 1 + _^..-^ + -J^_l (3) u

-^ n G)Ai G^?n u

^^^ o o o

Also from (CII-^) and (III-5), and (IH-l^) and (III-8)

n R + D + cm

(n +(m, +S) f

q = cm2 ^

g

3

(5)

(R + cm )

where a refers to the electron-ion and ion-ion recombination coefficient. In

the following we ignore D + cm in comparison with R, thus strengthening the

inequality (3). Thus:

u , P- u. o^ / R + ^o \^

u ^T. d 3. ion 3 f^\ o o o * '

We now make some estimates of — for various levels of the D region

o perturbed as in Section V, Fig, 6 , We let the 3-body coefficient be

-30 6/ -7 3 -3

K = 2 X 10 cm /sec, oc = 5 x 10 cm /sec, G = 2 x 10 and the values of

n, V, u and [o J are obtained from Table VI-I. The important parameters are

provided in Table I,

82
TABLE I

Parameters to use in (6)

u

Altitude (km) Gv sec" Vo sec ^i

u

50 1.32 X 10^ kk 82.5 275

60 3.8 X 10^ h.3 90 300

TO 8.7 X 10^ .39 108 32i^

Using these parameters in (6) we can olDtain upper "bounds on — as o in Table II.

TABLE II

Upper bounds on average energy of electrons

Altitude (km) u

50 1.092

60 1.059

70 1.031

The deviations from 1 are negligible and the assimption that the electrons

are at the same temperature as the background gases (even for a seriously

perturbed D region) are justified.

It is of interest to compare the result (6) with that of Sears [1963J •

In his work he did not have to include the ionization process since he dealt

with a much less severely disturbed ionosphere than we had. His formulation,

equivalent to Eq. (6) but of different form, contains the ratio of the 2 -body

attachment rate to the energy loss rate. We have the ratio of the 3-body rate

83
to the energy loss rate. In our derlvd,tion this ratio comes from the assumption

that the main source of n_ is the 3-T3ody process. If the n did, in fact, come

from the S-body process then we would have to use that rate in (6). We can

resolve this matter "by the following: The maximum 2 -"body rate for electrons

k -1 f X at 70 Im altitude is 10 sec at 3 eV and is negligible "below 1 eV (see Fig. 2 ).

Now, the energy relaxation rate for these energetic electrons ( » 1 eV) is more

than Uo times as great [Frost and Phelps, 1962! . Thus the nascent electrons

quickly decay in energy below 1 eV with very few of them undergoing any 2-body

attachments. Thus the n by defavilt comes from the 3-body process. Sears, in

his computations, seems to have ignored the electron energy dependence of the

energy relaxation rate, and thus obtained excessively high average energies.

Acknowledgement '

I wish to thank Dr. John Brooks for preparing the nonlinear propagation programming

which resulted in Figs. 7 through 17, and Mr, Wayne Ganzell for programming

Eqs. HT-l through III-3, as well as Dr. John H. Gardner and Dr. S. Altshuler

for many valuable discussions.

8i^
REFEIRENCES

Agard Ionospheric Committee Proceedings (December I961) Fifth Meeting, Athens,

Greece, J. Atmos. and Terr. Phys. 23

»

Allis, W. P. (1956) Handbuch der Physik, 21, Springer Verlag, Berlin, Germany.

Altshuler, S. (Aug. 15, I963) Effects of inelastic collisions upon electrical

conductivity and electron heating in the lower ionosphere,

J, Geophys. Res. 68, No. 16, kjOJ-kTi^.

Anderson, J.M. (Sept. 19^1) Electron-ion recombination in low temperature

gaseous plasmas. General Electric Research Laboratory. Rept.

No. 6I-RL-2817G.

Armour Research Foundation Project Suggestion (1961) Investigation of electron

removal and propagation in the upper atmosphere. No. 61-I26 AV.

Barrington, R. E. and E. Thrane (January, I962) The determination of D-region

electron densities from observations of cross modulations,

J. Atmos. and Terr. Phys. 2jj-, No. 1, 31-^2.

Biondi, M. A., (Nov. 20-21, 1958) Effects of high altitude nuclear detonations on the propagation of electromagnetic waves. Paper presented at

meeting at AFCRL.

Burch, 0. S., S. J, Smith, L. M. Branscomb (Oct. 1, 1958) Photodetachment

of ", Phys. Rev. 112 , No. 1, 171-175.

Chanin, L. M. , A. V. Phelps, M. A, Biondi (Apr. 15, 1959) Measurement of the

attachment of slow electrons in oxygen, Phys. Rev. Letters 2,

No. 8, 3^^-3^6.

Chapman and Cowling (1939) Mathematical Theory of Non-Uniform Gases , Cambridge

University Press, Cambridge, England.

Grain, C. M. (April I961) Ionization loss rates below 90 Ian, J. Geophys. Res. 66

No. h, 1117-1126.

85
Grain, C. M. and P. Tamarkin (Jan. I961) A note on the cause of sudden ionization

ancmalies in regions remote fron high-altitude nuclear "bursts,

J. Geophys. Res. ^y No. 1, 35-39.

DASA Report 1229 (May 1, I961) Electronagnetic "blackout guide, 1, General.

Electric TEMPO, DA U9-IU6-XZ-O38.

Fite, W. L., J. A. Rutherford, W. R. Snow and V.A.J, van Lint (Oct. 11-13, I961),

Mass spectrometric studies of afterfLows, Paper presented at

iktb. Annual Gaseous Electronics Conference.

Frost, L. S. and A. J. Phelps (Sept. 1, I962) Rotational excitation and momentum

transfer cross sections for elections in E and N from transport

coefficients, Phys. Rev. 127, No. 5, I62I-I633, p. I629, Fig. J.

Ginzburg, V. L. and A. V. Gurevich (Sept. -Oct. I960), Nonlinear phenomena in a plasma located in an alternating electromagnetic field,

Soviet Physics Uspekhi 3, No. 2, 175.

Glasstone (ed.) (April I962) The Effects of Nuclear Weapons , USAEC, U. S.

Gov't. Printing Office, Washington, D.C.

Gordon, W. C. and L. M. LaLonde (Jan. I961) The design and capabilities of an

Ionospheric radar probe, IRE Trans., AP-9, No. 1, 17-22.

Handbook of Geophysics (1957) 1st Ed., Geophysics Research Directorate, AFCRC,

p. 5-79, Figs. 5-h2,

Kane, J. A. (Dec. I96I) Re-evaluation of ionospheric electron densities and

collision frequencies derived from rocket measuarements of

refractive index and attenuation, J. Atmos. and Terr. Phys. 23,

338-31^7 (p. 3i^5)

.

Karzas, W. J. and R. Latter (May I963) Air conductivity produced by nuclear

explosions. Rand Corp. RM 367I-PR.

Kasner, W. H, W. A. Rogers and M. A. Biondl, fOct.11-13, I961) , Electron-ion re- combination coefficients in nitrogen and in oxygen. Paper presented at

ik-tb. Annual Gaseous Electronics Conference

86
Klhara, T. (Jan. 1952) The mathematical theory of electrical discharges in

gases, Revs. Mod. Phys. 2h, No. 1, i<-5-6l (p. k9)

,

Margenau, H. (May 1 and 15, 19^4-6) Conduction and dispersion of ionized gases

at high frequencies, Phys. Rev. 69, Nos. 9 and 10, 508-513.

Minzner, R. A., K.S.W. Champion, and H. L. Pond (Aug. 1959) The ARDC model

atmosphere, 1959; Geophysics Research Directorate AFCRC,

AFCRC-TR-59-267.

Mitra, S. K. (1952) The Upper Atmosphere , 2nd Ed. , The Asiatic Society,

Calcutta, India.

Molmud, P. (Feb. 1956) The energy of electrons in a gas in an arbitrary electric

field, Ramo-Wooldridge AP-58.

Molmud, P. (Dec. 1958) The modification of the lower ionosphere hy groundhased transmitters. Space Tech. Labs., GMOO-4730-00056.

Molmud, P. (Feb. 1959a)The modification of the lower ionosphere by groundbased radio transmitters, presented at the AFCRC meeting on The Artificial Modification of the Ionosphere, 22 January, 1959,

Space Tech. Labs., TR 59-0000-00592.

Molmud, P. (Apr. 1, 1959'b)Langevin equation and the ac conductivity of

non-Maxwellian plasmas, Phys. Rev, 11^ , No. 1, 29-32.

Molmud, P. (Oct. 1959c)The nonlinear propagation of strong electromagnetic

waves through weakly ionized media. Space Tech. Labs.

TM 59-0000-00397.

Molmud, P. (Jan. 1962a)Graphs of the real and imaginary parts of the effective

collision frequency for weakly ionized air. Scientific Report

No. 1, AFCRL 62-55, Contract AF 19{6ok)-Q8kh, Space Tech. Labs.

862 5-6001 -RU-000.

Molmud, P. (July 196213 )Altering the electron density in the D region, a new

tool for ionospheric D-layer research. Scientific Report No. 3*

AFCRL 62-533, Contract AF 19(6oU) -88U-, Space Tech. Labs.

862 5-600 5-RU-OOO.

87
Molmud, P. (Oct. 1, 1963) Altering the electron density in the D region: a new tool for ionospheric D-layer research, J, Geophys, Res. 68,

No. 19, 5^3-5^51.

Munick, R. J. and R. T. Folk (March I960) Electron capture in the ionosphere Ijy

dust particles, Aeronutronics, AFCRC-TN-6O-I86.

Navrocki and Papa (Aug. I96I) Atmospheric Processes , Geophysics Corporation of

America, Bedford, Mass., may be obtained from U. S. Dept. of Commerce.

Nicolet, M. and A. C. Aikln (May 1960) The formation of the D region of the

ionosphere, J. Geophys. Res. 65, No. 5, ikS^-lkQS.

Obayashi, T., S. C. Coroniti, and E. T. Pierce (May 23, 1959) Geophysical

effects of high-altitude nuclear explosions. Nature 183, lh'j6-lk'J&,

Phe2^s, A. V. (Oct. I960) Propagation constants for electromagnetic waves in

weakly ionized, dry air, J. Appl. Phys. 31^ No. 10, 1723-1729.

Reid, G. C. (Dec. I961) A study of the enhanced ionization produced by solar

protons during a polar cap absorption event, J. Geophys. Res. 66y No. 12, U07I-J+O85.

Sears, R. D. (Sept. 13, 19^3) Contribution of energetic photoelectrons to D-region

nonequilibriimi electron temperatures, J. Geophys. Res. 68, No. I8,

5135-51^2.

Sen, H. K, and A.A. Wyller (Dec. I960) On the generalization of the AppletonRartree magnetoionic formulas, J. Geophys. Res. 65, No. 12,

3931-3950.

Shkarafsky, J. P. (1961) Values of the transport coefficients in a plasma for any

degree of ionization based on a Maxwellian distribution.

Can. J. Phys. 32> 1619-I703.

Stanford Research Institute (no date) Parabolic Antenna Nomograph.

88
Nonlinear Ionospheric Two -Plasma Processes

Professor Joseph E. Rowe

Electron Physics Laboratory, Department of Electrical Engineering,

The University of Michigan, Ann Arbor

The subject of ionized stream interactions and vhistler

modes has received considerable attention by Gallet and HelliweU.,

Rydbeck, etc . , in which it was postulated that the low -frequency

emissibns of the exosphere were due to a traveling -wave -type interaction between drifting ionized streams and whistler modes. Several

linear theories have been developed to describe the possible interactions in terms of space -charge waves, cyclotron waves and magnetoionic waves. The solution of the system secular equation yields the

wave eigenvalues but no information on the energy exchange process

or on the particle distribution functions.

In this paper the nonlinear plasma -electromagnetic wave

interaction process (multi -dimensional) is examined in a Lagrangian

framework for both single -stream and multi -stream configurations.

It is shown that nonlinear ionospheric processes may be described

by simply generalizing previous microwave electron beam device

theories. Arbitrarily oriented static magnetic fields are included

and the general theory applies for both oa /co < 1 and cjo /a> < 1.

Many ionospheric processes are multi -stream in nature and thus a

89
fiaite -te.nperature plasrn.d is considered joid an arbitr^rj' velocity

distribution function may be specified.

In addition to the steady-state solutions given by the above

theory, it is possible to obtain the transient solution by solving

the nonlinear collisionless Boltzmann-Vlasov equation by an expansion method developed by Dolph and Lomax at the University of

Mchigan. At present this theory is limited to one -dimensional

phenomena but yields an accurate description of hot plasma phenomena and may include the effects of collisions.

90
INTRODUCTION

In the past few years a great deal of work has been done on

both natural and man-made VLF emissions in the ionosphere. Gallet,

Helliwell and Rydbeck have made theoretical studies on the basis of

the existence of multi-stream electromagnetic -wave interactions like

those taking place in klystrons and double -stream microwave amplifiers.

Their analyses are linear (small amplitude) and yield only information

on the dispersion characteristics of the combined stream-wave system.

It is interesting to note that in addition to both forward and backward electromagnetic waves two space -charge waves, two cyclotron waves

and two magneto -ionic waves can propagate in the system, yielding an

eighth-order system.

Since these authors have neglected second-order and higher terms

in their analyses* no information has been obtained on the energy exchange processes. Furthermore, raulti -streaming effects and finite

temperature electrons and ions have been neglected in order to simplify

the analysis. The complete understanding of Whistler mode generation

and propagation in the ionosphere requires a detailed nonlinear theory

91
which accounts for such effects as multi -streaming, large -amplitude

oscillations, finite- temperature plasmas, collision effects, and

energy exchange between the plasma (or electron stream) and any

electromagnetic wave propagating medium. The two nonlinear methods

to be discussed briefly in this i^aper yield such information through

a ballistics treatment of the problem and through the solution of the

nonlinear Boltzmann transport equation. The former method is developed along similar lines to the nonlinear analysis of the klystron

and traveling -wave amplifier taade by the author and the latter method

is due to Professors C. L. Dolph and R. J. Lomax at the University of

Michigan .

LAGRMGIM BAIXISTICS ANALYSIS

The electron stream-plasma interaction is depicted in Fig. 1

wherein the model is composed of two interpenetrating streams surrounded by either an infinite conductivity drift tube or a wave propagating medium. The usual electron stream is virtually a plasma since

it is collisionless and has a simple velocity distribution such as a

si^uare or Maxwell-Boltzmann distribution and thus may be analyzed on

the same basis as a plasma. The drift tube is considered to be cut-off

for all frequencies carried by the plasmas so that there is no electromagnetic wave propagation in the region between the streams and the

tunnel.

92
oH

f>4

93
The above system Is in general a multi -dimensional one and may

Include axial and radial magnetic fields as well as finite -velocity

distributions in the streams. For simplicity only a one -dimensional

system will be described here although finite-temperature effects

are included. In order to assure one -dimensionality of the system,

an infinite axial focusing field (magnetic) is assumed so that

u) /cD -» 00 .and thus both signal frequencies and stream plasma fjrequencies are well below the cyclotron or gyrofrequency.

In the Lagrangian method we integrate from some initial plane

(z = O) along particle trajectories^ summing all forces at each displacement plane. In the event that there is a wave propagation

medium present it may be treated on an equivalent circuit basis derived

from Maxwell's equations. From the 7 x E equation the following

second-order differential equation for the electromagnetic wave

equivalent voltage variation with distance may be written

where

V(z,t) = the wave equivalent voltage amplitude,

p(z,t) = the r-f charge density in each stream,

v^ = -pzz ) the equivalent e-m wave phase velocity, and

o

Z «= vL/C , the equivalent e-m wave impedance.

9^
The plus sign on the right-hand side is used for forward -wave interaction and the minus sign for backwai\i-wave interaction. In many

instances the fermeability and permittivity of the medium determine

both V and Z . o o The continuity of entering charge

p^dz^ = pdz (2)

and the Lorentz force equation for the one -dimensional system

dv I I dt = 1^' E + E

c-z sc-z

(5)

where

E = the axial component of the e-m wave field, and

c-z E = the axial component of the Coulomb force field,

SC ""Z

complete the system description.

After a considerable amount of mathematical manipulation the

following particle force eviuations evolve

:

95
^u (?;,$ .^ -e q:(1 + ii')

^ °J

^

2(1 +b^2

4^ cos $ (y,0 ,) - A(y/ - - ^|^ ") sin <t (y,$ ^^ dy 1 ^

-^ ' oJ

^ "^ \ a dy y i ^

-

' oj

2n

F dO'

11 oj

(1+bf ^ ^^/o/q [1+Q;u,(y,$ n

€.€2(1 + n')

(1 +b)^(l - T)')

'^iiiP2

o) q:

gi oj

o [1 +a u^(y^*oj^]

M

and

2^ - ' oj ' ^,

-t ci;(l+ Ti')

2

2(1 - b)2

d—A(v> -ij.— cos dy * (y.<5 J . A(y/ - - ^1^ ^ sin (y,4) .' gVJ^ Qj ^'^ \ q; dy y 2

'''' OJ

(1 - bV

20) ^ 2

CO J!

2jt ^F d$" 22 01

[1 +Q! u^(y,<5^j)]

(1 - b)^(i + V)

Pi

0) Ot

2jt F d$' 12 01 o [1 + a

^^(y^^oj^^ (-:>)

96
where

b

^^ 1 - u /u

^ 02 Ol

1 + u /u

* 02 01 , the

strean relative velocity difference,

A

1 - K\ 1 + 'n /i, I 2 1

the relative .Tiase ratio,

2 Z I V (z,t) = V(y,0) = Re

q:o o ./ v -j^

: A(y) e

the e-m wave voltage,

y = X = O5coz/2u , normalized distance,

^(z,t) = oi ( ^^ - t ) - 9(y), charge group ph^ise variable

,

dt == %i f^ -"'^ ^1 ^y^^oj^^'

dz A

dt u [1 + a u^ (y,* J], 02 2 ^ ' oj

u = average velocity of stream - 1,

u = average velocity of streajn - 2,

q; = 2C = V /V . , depth of modulation,

g' o

"^ A= stream radian plasma frequency.

97
The first terms on the right -heind side of B^s. Ovand^^represent

the e-m wave field and the integral terms the Coulomb fields. The

weighting functions ^^^^F and ^ F are shown in Figs. 2 and 3 vs. the

relative phase positions, * - **, The ^^^^F weighting function is nearly

exi)onential as expected and does lead to scane computational difficulty

when $ - *' ->0. This may be alleviated to some extent by utilizing

a large nimber of charge groups in both the stream and the plasma.

The above nonlinear equations are solved as an initial value

problem on a high-speed digital ccMuputer. Some representative

solutions for a double -beam system (c = g = - 1"^ plus an e-m wave

are shown in Fig. k. Here the fundamental r-f current densities are

normalized with respect to the d-c current in either stream. The

distance variable y may also be expressed in terms of the characteristic

Debye length as follows I

>r2 a)_ ^ ^ /

The efficiency of energy conversion to the e-m wave is shown in Fig. 5

as a function of both the plasma frequency and the mean velocity of the

streams relative to the e-m wave.

1 + ^ = ^

2 V

o

98
oII jo" in 1^ 00 o o d

01 II II

II

II o o o \ v. Vv — ^ CM lO CD J3 JH Si

in

CVi

^.H

99
IDO ^

•1

m f^ 00 ^ o d o

'T^o

n

IIo

"-"v \ ^

tvi fO

m ja X> Si

d

ID5.o od

100

BHEHO

PmO

COH

1^

100
PIO. i*^ DOUBLE-BEAM AND dRCUIT WAVE. TOHDAMENTAL COMPONENT OF

BEAM CURRENT VS. DISTANCE AND SPACE CHARGE. (C = 0.1,

b = 0.1, p = b, € = e = -1, I'

1 3 101

0, n' = 0, = -50)
PI P2

0.3

FIG, 5 DOUBLE -BEAM AND CIRCUIT WAVE. EFFICIENCy VS. o) /cii

AND p. (C = 0.1, b = 0.1, € = € = -1, I' = Tj' X 0,

1 2

t^ = -30)

102
A value of p « 1 corresponds to an excess stream velocity of approximately 10 percent.

The harmonic current components for a simple two -stream (electron

beam -plasma electrons) interaction are shown vs. the distance variable,

X*y = jtQ!N ,in Fig. 6 where the normalization has been made with

s respect to the individual stream currents and the mean stream current.

The maximum value of i /l^ .is given by the ballistic value

i—- = 2 J (1.8^) = 1.16 , o ^

The interaction phenomena involved when an electron stream is

coupled to the ions of a plasma may be studied by simply setting

€ = -1 and € = + 1. We aire primarily interested here in an inter1 2

action process in which there is no e-m wave. The results for varying

b, n* and (u /cd are summarized in Fig. 7 where the stream currents ^

Pi ' have been fconsidered equal. We note that i /l-j. can rise considerably

above the l.l6 value emd furthermore that the results are critically

dependent upon b, oj /oOjand r\*

.

Finite -temperature electron and plasma effects are readily

treated by giving the charges initial velocity distributions according

to Fig. 8. The velocity spread is denoted by S = (v -v Vu and

indicates the 'deviation from the mean. When S is small there is little

effect on the results as would be expected. Numerous results on the

effects of finite S are summarized in Fig. 9 for a two -electron beam

Interaction.

103
tn o

H

o

II

II ^3

o

II

(=•

II

s

vo

I/^!

104
I „.

(MO

cC

.0,

x^^s^^i^^'^^O'T" '"^"^^

b=0,JL,V=0.5

b=O.I,^ 7,-0.5

I „•

J-02

b=O.I,-p-,77'=0.8

b=O.I,-^,77'--0.5

•01

0.2 0.3

FIG. 7

(Upi /CU= OJpg/CU

HSAM-PLASMA INTERACTION. FUNDAI1ENTAL CURRENT AMPLITUDE

VS. 0) /co, b AND T)'. (e = -1, e = +1, a = 0.2, I* = O)

P 1 2

105
m;

u

THE NUMBER OF

CHARGE GROUPS

PER VELOCITY

CLASS.

u

FIG. a VELOCITT DISTRIHTTXCirS.

106
\.d ^a»0.2

(without vel. w group)

1.0 " / \

0.8 - \

/ a=0 a =0 a =0 a=0

3-0.6 - / s, =82=0.4 8, = $2=0.2 81 = 82=0.2 81=82=0.1

/ N|=N2=4

^

N,=N2=4 N, = N2=8

/

N, = N2 = 4

0.4

0.2 // I

/y

\ r/^Tv

i

^ ^

a=0-^ y^ Si=S2 = 0.l

"'^ N,=N2=8 \y s, a = = 82=0.4

= N2=8

^v^

.1 1 1 1 1

1

0)1, /lo,

0.5 1.0 1.5 2.0 2.5 3.0 3.5

y

-IS 0.6-

FIO. 9 DOOBIS-HEAM nrTERACTlUfl . VELOCnT DI3TEIBUTI0H

CHARACTERISTICS, (t - 0.1, a)^/(u ' '^^J'"

" °'^'

e - e - -1, I' - n' - 0)

1 2

107
NONLINEAE BOLTZMAIiTJ EQUATION

An alternate approach to the solution of the nonlinear

interaction problem investigated by Professors Dolph and Lomax at

the University of Michigan involves the solution of the BoltzmahnVlasov transport equation by the method of expansions. They solved

the following coupled transport equations

bf ^f bf

at dx m. ^v

and

^f df hf

ot dx ^± ^v

where

f = the particle distribution function,

X 00 d X* 00 E = l- (.f^ - f^) dv dx'.y y y 1^ ^^i-^e^ ^^ ^' ^"

O ^ \J U Q O -co O O -00

V

- J

+ E ; the electric field for a simple diode,

d = diode spacing.

The solution method involves the expansion of the distribution

function in terms of orthogonal i>olynomials in the velocity variable.

Half-rajGge expansions are made in terms of Laguerre polynomials LT (v)

108
which are orthogonal with respect to the weighting functions

t +v

(xr{y) = e . The actual details of the method involve working

with the moments

M^ =

I v^ f dv (Q)

M^ = / v^ f dv . (9)

So far this method has been applied to the single and two -stream

diode to obtain both the transient and steady -state solutions. The

results agree well with other results and indicate promise for the

method on more general problems. In the electron -ion problem, the

electrons come to equilibrium in a considerably shorter time than do

the ions and therefore the solution time must be scaled to the convenience of the ions.

CONCLUSIONS

It has been established that a nonlinear theory is required to

analyze the generation and propagation of VLF ionospheric emissions.

Previous theories have been linear and on^ yield information on the

system propagation constants generally neglecting finite -temperature

effects, multi-streaming.and collisions.

One of the nonlinear theories constitutes a Lagrangian ballistics

solution of the equations of motion and includes all of the above

109
effects . The effects of colliBions are incorporated by including an

electric field term vhich is developed from the appropriate collision

model. The other ncailinear theory yields both transient and steadystate solutions and allso includes the above effects. Numerous steady*

state solutions obtained by the Lagrangian method were presented and

the results indicated that the methods are applicable to the analysis

of other ionospheric two -plasma phenomena.

110
List of Figures

Figiire

1 Eeam-Plasma-Circuit Interaction Configtiration. 93

2 Si>ace-C2iarge -Field Weighting Function F. 99

^ Space -C5i€urge -Field Weighting Function F. iqo 12

h Doutle-Beam and Circuit Wave. Fundamental

Component of Beam Current Va . Distance and

Space Charge. (C =- 0.1, t = 0.1, p = b,

£ 12= € = -1, I' = 0, n' - 0^ t^ = -^'0) 101 o

5 Double -Beam and Circuit Wave. Efficiency

vs. <i>^<ii And p. (C = 0.1, b = 0.1, e =

e = -1, I' = n' » o> ^^ = -50) 102

2 o

6 Double -Beam Interaction, (s = e ^ -1?

a = 0.2, I' = T)' = 0, to /to = CD /(D = 0) 104

7 Beam-Plasma Interaction. Fundamental Current

Amplitude vs. <d /o^, b And t

' . (c = -1,

- ^ +1, a = 0.2, I' = 0) 105

2

8 Velocity Distributions. 106

9 Double -Beam Interaction. Velocity Distribution

Characteristics, (b = 0.1, to /o «

^t)2/*^ * ^'^•'

= -1, I' = T]' = 0) 107

1 2

111
RBFERENCas

1. Rove, J. E. "a I^rge -Signal /^Jialysis of the Traveling-Wave

Amplifier: Theory and General Results", Trans . PGED-IRE,

vol. ED-5, pp. 59-57; January, 1956.

2. Dolph, C. L.J lamaxj R. J.j "On the Nonlinear Theory of

Multistream Electronic Devices", Proc. _of the Conference

on Wave Interaction and Dynamic Nonlinear Phenomena in

Plasmas , The Pennsylvania State University; February

h to 6, 1965.

112
CONCLUDING REMARKS

E, K, Smith, Jr.

National Bureau of Standards

This part of the program has been set aside for final comments.

What I will do is to make some general remarks myself and then ask

Dr. Donald T. Farley to present a short technical summary of the

meeting.

On behalf of Dr. Menzel and myself I would like to thank all

of the delegates for their contributions to the success of this meeting.

We are particularly grateful for the contributions from our colleagues

from overseas Professor V. A. Bailey, a true father of the field, has

come all the way from Australia despite poor health. His former

student, Dr. R. A. Smith has come a similar distance and has made

it possible for an American audience to appreciate the excellent work

going on at the University of New England, Professor Cutulo has

surmounted the language barrier to present the very interesting work

he has carried on for nearly two decades at Naples. We also owe our

thanks to Dr. DeBarbieri who has given us a good insight into the fine

work of Professor Caldirola and himself at the Universita Di Milano.

Finally we need to thank Dr. Belrose for his vigorous participation in

the Conference. It would indeed have been awkward to leave the important work which has been going on at Ottawa luirepresented at this

Conference.

On your behalf^ Dr. Menzel and I would like to convey to

Professor Rydbeck our regret that he could not be with us and to wish

him a speedy recovery.

I would like to acknowledge the pleasure I have had in working

with Dr. Menzel as Co-chairnaan of this Conference and also the fine

efforts of Robert T. Frost and his colleagues on the Local Arraaigements

Committee.

Finally, Dr. Menzel and I would like to thank the Advisory

Com.mittee, made up of Dr. Little, Professor Layzer, Dr. Wait,

George Jacobs, and Roger Gallet, for their assistance with the technical

program and publication plans, and the Voice of America for their

assistance in helping to defray the expenses of this Conference.

113
CONCLUDING REMARKS

Conference on Non -Linear Processes

D. T. Farley, Jr.

National Bureau of Standards

The effects that have been discussed at this conference can be divided

into two broad categories, in one of which collisions are important, while

in the other they are negligible. In the first group the non-linearities

are created by temperature variations in the medium caused by the electromagnetic waves. This group can be subdivided into weak effects, in which

the temperature is perturbed only slightly, and strong effects in which the

properties of the medium are changed drastically.

Research has been carried out in the first of these subcategories for

many years. Cross -modulation, or the "Luxembourg effect" , has been used by

several groups to study the D and lower E regions of the ionosphere. The

general features of the theory of this effect are quite well understood.

It has always been difficult, however, to interpret the measurements vm- ambiguously and obtain from them profiles of electron density and collision

frequency as a function of altitude. One of the important results demonstrated at this conference was that it is impossible to uniquely determine

these quantities from cross -modulation measurements, even when the data are excellent. An infinite number of solutions, many of which appear quite

reasonable, can apparently be made to fit the data. Without an independent

measurement of the electron density or collision frequency, it appears that

measurements of the Luxembourg effect are of limited use as a diagnostic

tool for the lower ionosphere

.

When we turn to the subject of stiong heating effects, the prospects

appear more hopeful. It is now feasible to radiate sufficient power to

substantially alter the characteristics of the ionosphere, particularly if

the frequency of radiation is near the electron gyro -frequency. It has

been appreciated for some time that the ability to raise the temperature

of the ionosphere by an order of magnitude or more would be very useful in

studying the various chemical and excitation processes in the ionosphere.

114
From rather simple calculations one can arrive at a qualitative understanding of what -will happen. However, as we have heard here, such calculations cannot be expected to give even semiquantitative answers to such

questions as how much power is necessary to produce an artificial airglow.

The magnitude of most of the excitation effects depends very critically on the shape of the distribution function of the electron energies, as well

as on the mean energy. It is the very small minority of electrons with

very high energies that we are most interested in. In general the distribution function is far from Maxwellian, and so any calculations which are based on the ass\imption of such a distribution are likely to be seriously

in error. We have heard reports at this conference of some detailed

n'umerical calculations of the distribution function that show great promise

of helping us to understand the details of the chemical and photochemical

processes in the ionosphere.

Such calculations have been greatly aided by recent laboratory measurements of many of the pertinent cross -sections for the various reactions.

These measurements are now being made at energies only slightly higher than

those attainable in the ionosphere using very high power transmitters.

To summarize, then, the situation regarding collisional effects in the

ionosphere, it appears that the most productive future research may be

largely in the realm of very high power experiments. Such experiments,

coupled with laboratoiy measurements and detailed calculations, should

give us a much better understanding of the chemistry of the lower ionosphere.

The second class of non-linear phenomena we have considered here do not

involve collisions; they can be called coherent non-linearities. These

effects occur when charged particles move at approximately the same velocity

as the wave. Then the waves and particles can interact strongly, causing

either the waves to grow or the particles to accelerate. Since there are very few relativist ic particles in the ionosphere, we are concerned mainly

115
with slow waves, such as VUF emissions, magnetohydrodynamic and electrostatic waves, etc. With the possible exception of the resonances observed

by the Alouette topside sounder, our understanding of these effects appears

to be very limited at present. It is becoming clear, in particular, that

the variety of VLF emissions that are observed are doubtless due to a variety of complicated interactions occurring in the exosphere. However,

the details of these processes are not at all clear at present. This is

another area in which laboratory measurements may eventually be helpful,

since many similar effects occur in laboratory plasmas. Hopefully there

may be more interaction in the future between these two fields of research

than there has been in the past.

116

r I

1





I