national Bureau of Standards

library. N-W. Bldg

DEC 7 1965

^ecUnlcui ^iote ^o. 325

NUMERICAL SIMULATION OF IONOSPHERIC

WAVE INTERACTION EXPERIMENTS

THOMAS M. GEORGES

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical cMote.5<25

ISSUED October 25, 1965

NUMERICAL SIMULATION OF IONOSPHERIC

WAVE INTERACTION EXPERIMENTS

Thomas M. Georges

Institute for Telecommunication Sciences and Aeronomy

Environmental Science Services Administration

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: 304
V
ABSTRACT

The Fejer [1955] pulse interaction experiment is simulated numerically

by solving, in an iterative fashion, the electron energy balance and

magnetoionic propagation equations over a height-time "matrix" representing a disturbed ionosphere. The complex interdependence of

physical variables is more accurately represented by including "selfdistortion" effects on the disturbing pulse, and allowing electron

collision frequency and the energy loss coefficient (G) to vary with

electron energy and time. Provision is made for nonrectangular disturbing pulses as well as finite-length probing pulses. Phase as well

as amplitude interaction effects are calculated.

Absorption and phase shift computations are based on the AppletonHartree or Sen-Wyller complete (arbitrary angle of propagation)

formulations, but the insertion of more sophisticated developments in

interaction theory, including electron density perturbations, should be

possible without appreciable increase in program complexity.

Representative calculations demonstrate the effects of variations in

experimental parameters , perturbations of ionospheric conditions,

and various interpretational assumptions.

The numerical approach offers advantages of simplicity and flexibility

to those engaged in design and interpretation of wave interaction experiments. Synthesis of D-region electron density and collision frequency

profiles, using the technique "in reverse," should prove superior to

presently used trial-and-error methods.

Key Words; Cross modulation, D region, ionosphere, Luxembourg

Effect, nonlinear propagation, wave interaction

111

Contents

Page

Abstract iii

1. INTRODUCTION 1

Z. LIMITATIONS OF EXISTING ANALYSES Z

Z. 1 The Experiment Z

Z. Z Analytical Interpretation 4

Z.Z. 1 The Av Term 5

Z. Z. Z The Sk /ov Term 7

P

Z.Z. 3 Assumptions 8

3. NUMERICAL SIMULATION TECHNIQUES 9

4. SOME SAMPLE CALCULATIONS 19

4. 1 Variations in Disturbing Pulse Length Z3

4. Z Variations in Frequency and Polarization Z4

4. 3 Dependence on Disturbing Frequency Z4

4. 4 Neglecting Self-distortion and Energy

Dependences of G, V Z7

4. 5 Dependence on Form at G(Q) Z7

4.6 Perturbations of the Electron Density Profile 30

4. 7 Disturbing Wave Self -distortion 30

4. 8 Use of the Generalized Theory 3 3

4, 9 Gyro-interaction Effects 33

5. CONCLUSIONS AND LIMITATIONS 36

6. REFERENCES 40
Illustrations

1. Height-time representation of the Fejer experiment

2a. Operation of the simulation program

2b„ Height-time-energy representation of a disturbed

ionosphere

3. Functional diagram of the simulation program

4. Reference phase and amplitude interaction "profiles"

5. Electron density profiles

6„ Collision frequency and time constant profiles

7. Effect of disturbing pulse length

8. Effect of frequency and polarization

9. Peak interaction vs. disturbing frequency (day)

10. Energy dependence of G

11. Interaction for various G(Q)

12. Interaction for bulges in N(h) profile

13. Self-distortion of disturbing pulse

14. Interaction using generalized theory

15. Peak interaction vs. disturbing frequency (night)

16. Collision frequency perturbations for gyro-interaction

Page

3

12

13

14

18

20

21

22

25

26

28

29

31

32

34

35

37

VI
NUMERICAL SIMULATION OF IONOSPHERIC

WAVE INTERACTION EXPERIMENTS

Thomas M. Georges

1. INTRODUCTION

The present deficiency in the state of knowledge of the ionospheric D region has revived an interest in the cross modulation or radio wave interaction technique for the measurement of ionosphere

electron density and collision frequency. Since the discovery of the

Luxembourg Effect in 1933, numerous researchers have attempted to

demonstrate the feasibility of utilizing cross modulation to obtain

accurate ionospheric profiles [Fejer et al. , 1955, 1959; Rumi el al. , 1958, I960, 1961, 1962a, 1962b; Barrington et al. , 1962a, 1962b, 1963;

Titheridge, 1962; Smith et al. , 1952, 1964], However, the availability

of very high-power radio pulse transmitters and sophisticated informationprocessing schemes has only recently indicated that sufficient experimental accuracy is obtainable. The necessity of using such high power

to disturb the ionosphere leads to an examination of certain approximations implicit in present analytical methods for the interpretation of

experimental results. It will be demonstrated that it is not feasible to

account adequately for the complex interdependence of physical variables

in a purely analytical fashion, and that numerical techniques provide a convenient and flexible means of interpreting ionospheric pulse interaction experiments. This work outlines a technique for the simulation

of pulse interaction using a digital computer, and shows the dependence

of measured quantities on experimental variables and some common

interpretational assumptions. A forthcoming paper will demonstrate

the application of this method to the synthesis of ionosphere profiles

from interaction data*
2. LIMITATIONS OF EXISTING ANALYSES

2. 1 The Experiment

The experimental technique for obtaining ionosphere profiles

using cross modulation is based on the familiar pulse interaction

method developed by Fejer [1955], A very high-power, ground-based

"disturbing" transmitter sends a short pulse of electromagnetic energy

vertically into the ionosphere, imparting momentary increases in

average electron energy as the wave is partially absorbed at each

height. This alteration in electron energy, through an increase in

average collision frequency with air molecules, results in a perturbation of the wave absorption and phase coefficients of the medium. If a second pulse from a lower-power transmitter is caused to traverse

the disturbed region in a downward direction (after reflection from a higher layer), in effect "meeting" the upward-traveling disturbing

pulse, a perturbation of the received amplitude and phase of this probing

pulse will occur. (The word "probing" is felt to be more appropriate to

this experiment than "wanted, " which originated in the terminology of

the Luxembourg Effect. ) This perturbation is characteristic of the

disturbed conditions in a narrow height range near the meeting height.

If this sequence of probing and disturbing pulses is repeated, with the

probing pulses occurring twice as often as the disturbing ones, the

relation between adjacent probing pulses will contain the perturbation

information which can be interpreted in terms of electron density and

collision frequency near the meeting height. It is seen from figure 1,

which illustrates the experiment, that the height of interaction can be

adjusted by varying the relative delay time between pulses, and that

height resolution depends principally on the length of the probing pulse.
UJ^ CO — a

hthe onshi C .H

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CO c

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13 h-o O 7h

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CD CM \ "O

CD

13

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CU 1 ^ 1 CO h ^ CU T1 w -H 4->

CD

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coCD

CO

CO

— CD

CO r—, 3 3 "O —

Q. Q. CD =5 > Q_

CU D)

c: "S

p

bJO -Q 'jil CD 5= Cu O 13 en 'jd

i_

O

Q_ CO i_ O

a.
The study of the "nonlinear" behavior of the ionospheric plasma

has recently expanded to consider possible enhancements of electron

density due to perturbations of electron attachment coefficients and

effects associated with compressibility of the medium such as the

generation of several varieties of plasma waves, as well as the more

familiar process involving alterations in effective electron collision

frequency. However, only the last effect will be considered here,

since the others are believed to be of secondary importance in the

D region.

2. 2 Analytical Interpretation

In the interpretation of this experiment, the pulse interaction

process is usually represented by an integration of the perturbation in

propagation constant over the height range of interaction:

h. dk

| ^V^ ""^rdv

Avdh d)

o

where T is the complex interaction coefficient [Ferraro, Lee, and

Weisbrod, 1963]

T = m + j T , a

J cp k is the complex propagation constant for the probing wave,

k = a + jp

P P P

V is the electron collision frequency with neutral molecules,

h. is called the interaction (or "meeting") height of the two

pulses.

In terms of experimentally measured quantities,

AA

a A

often called the "modulation depth, " and
T = Acp (radians) (2)

where AA and Acp are the received probing pulse amplitude and carrier

phase perturbations as shown in figure 1. T has been defined with a plus sign in (1) in order to yield positive T for a Acp corresponding to

phase advance . Thus T is negative for frequencies above the gyrofrequency.

2. 2. 1 The Av Term

The collision frequency perturbation, Av, depends on the intensity

of the electromagnetic field through the electron energy-gain process.

For the relation between the energy of electrons and their collision

frequency with air molecules, use is made of laboratory determinations

of collision cross sections for air in the energy range of interest.

Experiments of Phelps and Pack [1959] have shown that, to a good

approximation,

Av = v Q - Qc Q

). 0)

The time dependence of v can be obtained by solving the

differential equation for electron energy balance,

dQ

dt

= v [e - G(Q - Q )] (4)

where Q is the electron energy = ak8,

Qo is the electron equilibrium (undisturbed) energy,

k is Boltzmann' s constant,

6 is electron temperature ,

£ is the energy gained from the wave by each electron per

mean free time,
a is a constant relating temperature to the energy distribution

G is the electron energy loss coefficient, the fraction of

excess energy lost per collision (usually determined

by laboratory measurements).

All of the variables in this equation are really probability

distributions, of which some statistical moment is usually used for

computations. Current practice is to use the most probable values,

in which case a = 1, and V is called v . m

The general solution of (4) is, for G and V nearly constant

within the interval Q to Q,

Q(t) = |, [l - exp (-t/T) ] + Q (5a)

for conditions of energy gain, that is, when the medium is absorbing

energy from the disturbing pulse, and

e(T ) t - t Q(t) - -^~ exp Q

-2- >j + Q ( 5b )

during the decay period after the pulse has passed. Here T = l/Gv

and T is the disturbing pulse length. Equations (5) are expressed in

d

terms of the height h of the probing pulse leading edge by substituting

h. - h

t = 2— (6)

where h. is the meeting height of leading edges,

c is the free -space propagation velocity.

In (4) and (5), G is usually treated as a constant equal to about

_3

10 , although various forms for the energy dependence of the loss
term have been considered in order to account for inelastic collisions

[Huxley, 1953; Altshuler, 1963; Benson, 1964]. The energy gain

term, e, is obtained by calculating the disturbing wave energy absorbed

at each height, per electron, per mean free time:

/-, x d d ,_. e(h) = —

v (7) Nv

where p is the disturbing wave power density at h,

P

d<

h> = -^- eXP(-Ai<*') (8)

N is the electron density,

CL is the disturbing wave power absorption coefficient at h,

d

P is the effective vertically radiated disturbing power.

2.2.2 The 3k /ov term

P

The term dk /Sv in (1) is obtained from some suitable expression

P

for the complex refractive index of a uniform ionized medium in a magnetostatic field. At this time, only the so-called Quasi-Longitudinal

approximations to the Apple ton-Har tree and the Sen and Wyller [i960]

generalized theories have been so treated [Benson, 1963, 1964]. When

the former theory is used, for example, an expression for the real part

of 3k /dV is of the relatively simple form

P

da (uu± a) f - v

2 P - AN (Q\

dv [((jo±u) ) 2 + \^] 2 {y}

1—1

where (D is the probing wave angular frequency,

a is the probing wave power absorption coefficient,

ir
uo T is 2TTf cos y » L, h

f is the electron cyclotron frequency,

Y is the angle of propagation with respect to the geomagnetic

field

_5 A =1. 062 x 10 in the m.k. s. system of units

N is the electron density

V is the effective collision frequency.

It is known, however, that such simplifications of the magnetoionic

theory are not strictly applicable under some conditions normally encountered in the ionosphere. Moreover, recent developments in the

theory of nonlinear wave propagation indicate that new expressions

based on a non-Maxwellian electron velocity distribution may be

required for waves of intensities which will be considered here [Layzer

and Menzel, 1965, and Budden, 1965]. For these theories, it may not

be possible to obtain expressions for dk /ov directly.

2.2.3 Assumptions

It should now be evident that with a proper substitution of (9),

(8), (7), (6), (5a), (5b), and (3) into (1) the integration (even numerically)

could become quite involved due to the complicated interdependence of

variables. In practice, at least one of the following rather limiting

assumptions is usually made:

1. The disturbing pulse does not undergo "self-interaction";

that is, no dependence of a in (8) on time or electron energy is allowed.

2. Electron energy rise and decay follow a simple exponential

behavior in time; that is, energy and time dependences of G and V are

neglected in computing e/G and T - l/Gv in (5).

3. The energy loss coefficient G is constant.
4. The disturbing pulse is rectangular, a condition seldom

encountered in practice.

5. Probing pulses are infinite sim ally short.

6. The difficulty of evaluating 3 k /3v for more complete

IT expressions for the refractive index restricts application to the

limitations of the approximate formulas.

Although the first three of these approximations are probably

not severe limitations under conditions of even moderately high disturbing

power, it is of interest to know within what disturbing frequency and

power ranges these approximations remain justified. Although the

applicability of 4, 5, and 6 clearly depends on the individual experiment,

it appears that in general it would be desirable to remove these

restrictions.

3. NUMERICAL SIMULATION TECHNIQUES

The nonlinearity of the analytical expressions which describe

the wave interaction process, and the complex interdependence of the

physical quantities involved, suggest a numerical approach to the

solution of practical problems. For this purpose a digital computer

program has been developed which simulates the upward passage of a

single disturbing pulse through a model ionosphere, its effect on the

medium, and the effect of the medium on a single downcoming probing

pulse. This approach has the advantages that all the approximations

mentioned above are unnecessary, and that a great deal of flexibility

is possible in the selection of experimental parameters and method of

treatment.

It will be assumed that some suitable expression for the complex

refractive index of a uniform magnetoionic medium has been chosen,

and that it is desired to apply it to a medium which can undergo only
small time variations in a wave period, and where only small space

gradients exist with respect to a wavelength. Under such conditions,

generally satisfied at high frequencies, the principles of ray theory can

usually be applied, wherein the medium is considered equivalent to a large number of space and time "cells" within which the macroscopic

properties of a uniform medium can be used. Overall nonlinear effects

can then be simulated by allowing the conditions in each cell to depend

on conditions in previous cells and on the disturbing wave power density.

Since the Fejer interaction experiment involves essentially

vertical propagation, only two dimensions need be considered -- height

and time. The disturbed medium can then be completely represented as

a two-dimensional height-time "matrix, " with each cell containing some

characteristic of the disturbed medium, such as electron temperature.

For ray theory to be accurate it is necessary that space and time

variations of the medium be small within each cell. Ionosphere

parameters do not change appreciably within height and time intervals

of 1 kilometer by 3 1/3 microseconds (the time for a wave to travel 1 kilometer in free space), and this convention has been adopted for convenience. It will be assumed that disturbing pulses are so much shorter

than effective recombination times that electron and neutral particle

density as well as gas temperature do not vary with time, and these

profiles, presently defined between 20 and 120 kilometers, are entered

into the program in tabular form in 1 kilometer steps. Electron

collision frequency is calculated from the electron temperature (energy)

and neutral particle density, rather than by a direct insertion of a v(h)

profile. The expression for the complex refractive index is entered as

a subprogram which can be varied independently, as are the dependences

of G(Q) and v(Q) which may be entered in functional or tabular form.

10
Experimental control variables are: disturbing and probing frequency

and polarization, disturbing power and pulse length, local gyrofrequency

and magnetic dip angle. The probing wave is represented by a series of

infinitesimal "delta" pulses meeting the disturbing pulse leading edge

at every 1 kilometer interval over the height range of interest. It is

assumed that interaction occurs in a nearly nondeviative region; that

is, the pulse group velocities effectively equal the free-space propagation

velocity, c.

It is also assumed that electron energy distributions remain so

nearly Maxwellian that the idea of an effective collision frequency as expressed in the Appleton or Sen-Wyller [l96l] developments is

applicable.

The operation of the program is most easily understood by

reference to figures 2a and 2b, which illustrate the wave interaction

region in the height-time plane, and figure 3, an operational diagram

of the program. Assuming only a free-space divergence of power

between the ground and hx , the disturbing power density, p , is

d

calculated for the first (lowest) height interval, hi , and the pulse

waveform is digitally represented by a one-dimensional array containing

values of p for each time cell. Any pulse shape can be defined at this

point, and the number of time cells required depends on the pulse length.

This waveform and the atmospheric parameters for the first height

interval are then used to calculate the time-dependent electron energy

for each time interval, using a form of (5) suitable for numerical

computations

,

Q(t+At) = Q(t) + [q + ^L - Q(t)][l - exp(^)] (10)

11
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Disturbing

wave

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leading

edge

/

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Disturbing

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trailing

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decay

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14
where At is a time interval sufficiently small that Q, G(Q) and v(Q) do

not change appreciably within it,

1

for ^ t ^ T ; zero otherwise, d

J. \L)

G(t) v(t)

e(t) A m in = 6. 242 x 10

18

• —^^

Nv

e is in electron volts,

P

d

N

/ 2

is in watts/ cm ,

. . -3

is in cm

a

d

V

is in nepers/cm,

. . -1

is in sec

Q =ak6 (h).

Using this expression in an iterative fashion, each time cell at

hx can be successively filled with a value for electron energy, from

which new values of G and V are calculated for the next time cell via

the G(Q) and v(Q) functions. In this way, the energy dependences of

these quantities are retained. The absorption coefficient, a., for the

d

disturbing wave and the absorption and phase coefficients, a and 3 , for the probing wave can now be calculated for each time cell at hi by

using proper values of N and v in the refractive index subprogram.

When enough time cells are filled after t = T that the electron energy

has nearly cooled to undisturbed conditions again, a is applied to each

element of the disturbing wave power density, so that when the 1 kilometer

spherical divergence attenuation is included, the disturbing waveform is

replaced by a new waveform for the next height interval, h.^ . This

waveform is of course slightly distorted from its original shape because

of the variations in a for each hx time cell. At h2 the new power density

waveform is applied to conditions at that height to fill its time cells as before with information on the propagation characteristics of the medium.

15
This process is repeated throughout the region of interest, so that the

resulting matrix covers the entire disturbed medium in height and time.

Knowing k in each height-time cell, the interaction contriP

bution of each cell can be calculated by subtracting the normal (undisturbed) value of k (k ) from the disturbed value:

P Po

6k = k - k . P P Po

The total interaction suffered by each element of the probing wave is

obtained by summing 6k over the diagonal set of matrix elements

Jr corresponding to each meeting height. The absorption encountered

outside the interaction region is the same for both a disturbed and an

undisturbed pulse, and is not included in the calculations since it does

not affect the interaction coefficient. To obtain the interaction information, it is not necessary to store the entire two-dimensional

matrix during computation, since it is known "in advance" which

meeting height corresponds to each time cell at each height. The values

for 6a and 63 may thus be stored in appropriate locations corresponding to each meeting height, immediately upon computation, and

summation carried out automatically when the next value for that

meeting height is encountered.

In the calculation of 6a and 6(3 at each height it is necessary

P P

to perform calculations for the complex refractive index for the value

of v in each time-height cell, and it is often necessary to subtract

values which differ by extremely small amounts. For larger differences

it is possible to calculate values of, say, k (v) and k (Vo) and subtract

p p

to obtain their difference; however, when differences become small, as

for very small interaction, this method is not satisfactory because of

the limitations of accuracy inherent in digital computers.

16
To calculate very small differences in refractive index, a

cal differentiation of k with re P

sufficiently small interval of Av that

numerical differentiation of k with respect to V is carried out over a P

dk Ak -H~-£ (11) dv Av

then

Ak

6k -—-£ 6v (12)

p Av

where 6v is the actual difference in v for which the value of 6k is

P

desired. Av is taken to be sufficiently large that an accurate value of

Ak / Av can be calculated, but still small enough that (11) is satisfied.

P

The principal output of the program is a tabulation of the phase

and amplitude r interaction coefficients, T and T for each meeting

a cp &

height where significant cross modulation occurs. These interaction

coefficients (figure 4) may be interpreted either as perturbations to

many very short probing pulses meeting the disturbing leading edge at

each height, or as representing the distorted shape of a continuous

probing wave received at the ground, such as in a cosmic noise or "sky

temperature" cross modulation experiment.

Among the "by-products" of the interaction computations are the

time functions of electron energy and disturbing power density waveform,

collision frequency, heating time constant and probing and disturbing

wave absorption parameters at each height, the approximate steady

state conditions which would result from a very long disturbing pulse,

and the total disturbed and undisturbed absorption for each wave.

These provide additional insight into the physical processes involved.

17
I

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co .<" d

1 UJ Pha

"rei

dist

1 cz>

cr> oo r

co lo

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18
4. SOME SAMPLE CALCULATIONS

The numerical technique just described has been used to perform a series of sample calculations in order to demonstrate the effects

of variations in experimental parameters and inter pretational assumptions

as well as to explore the general utility of the wave interaction experiment as a D-region diagnostic tool. In each of the results to be presented, a comparison is made between the amplitude interaction coefficient profile calculated from a set of "reference" conditions, and

those calculated under the conditions or assumptions in question.

Phase interaction effects are shown only for the reference conditions,

since they behave basically the same as the amplitude effects. The

Appleton-Hartree expressions for the refractive index are used for

most of these computations because of their simplicity; however, no

essential differences in the points to be illustrated results from the

use of more "complete" expressions. The reference conditions, chosen

somewhat arbitrarily, but approximating conditions believed to be experimentally optimum, are as follows:

7

disturbing vertically radiated power 10 watts

disturbing frequency 10 Mc/s (extraordinary)

disturbing pulse length 57 microseconds

probing wave frequency 3 Mc/s (ordinary)

The reference electron density profile (figure 5, curve C) is a "mean

midday" D-region profile for mid- latitudes resembling that in Knecht

and VanZandt [1964]. Neutral air density, M(h), and gas temperature,

8(h), profiles are from the ARDC Standard Atmosphere (1959). The

energy dependences of G and V are essentially the same as those of

Megill [1965], and represent the results of drift tube experiments of

Phelps cited therein. For small energy excursions these functions may

19
—

1 1

1 \ '

1

1 =

=

—

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o

CD

CO

>-O

UJZ>OUJ

a:

g

CO

o

o

o

CD

CO

I-

-z. <

co

-z. o

o

UJ

Figure 6.

60 70

HEIGHT, km

Profiles of electron collision frequency and heating time

constant resulting from the atmospheric parameters assumed

in these studies.

21
UJ

o

UJ

o

o

S PS

+- ij

« U -"- 5 > in

<

(Z

UJ

s

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UJQ

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22
be approximated by

v(Q) « 9.7 X 10" 8 Q* 9?8M

G(Q) -4.7X 10" 4Q"* 55

It should be emphasized that present results for G in particular are

subject to considerable doubt, due to the difficulty of making laboratory

measurements of the loss coefficient at such low electron energies;

however, these will be taken as the best available. Recent cross

modulation measurements [Smith, 1964, and Bourne, private communication] lend support to an approximately Q '" dependence of G in the

ionospheric electron energy range (. 02 to . 06 e. v. ). Various other

forms for the loss term in (4) have been proposed [Huxley, 1953;

Altshuler, 1963], but it appears that these can be accounted for by

just letting G = G(Q) as is done here. Figure 6 shows the collision

frequency and heating time constant profiles resulting from these

atmospheric parameters.

4. 1 Variations in Disturbing Pulse Length

Figure 7 shows the results of variations in the disturbing pulse

length. The rather unconventional scale used for T in this and

following figures is a convenient way of illustrating a quantity which

exhibits variations over a large range of positive and negative values,

although it may unduly accentuate small interaction effects. As would

be expected, longer pulses are more effective in causing heating in the

upper D region because of the longer heating time constants there; and

the height of zero interaction coefficient (integrated interaction) is

increased because of increased positive interaction at lower heights.

The shortest pulse length, on the other hand, provides little relative

emphasis on lower heights, its principal effect apparently being only a reduction in overall sensitivity. The relation between the "crossover"

23
heights of T and the height of actual zero interaction demonstrates the

difficulty of applying any simple relation to the crossover height to

obtain the height of zero interaction as an independent determination of

collision frequency there. It will be seen that this difference depends

on other experimental parameters as well. The height of zero interaction, however, does not significantly depend on disturbing pulse

length, since the heating time constant at this height is usually much

shorter than pulse lengths ordinarily used, and a quasi-steady state

electron energy is achieved quite rapidly.

4. 2 Variations in Frequency and Polarization

Figure 8 shows the magnitude of variations to be expected as a result of changing probing and disturbing wave frequency and polarization as well as the electron density profile to one representing typical

mid-latitude nighttime conditions. Of particular interest is the dependence on the disturbing frequency and polarization.

4. 3 Dependence on Disturbing Frequency

Figure 9 shows the values of the maxima of the interaction coefficients plotted against disturbing frequency. Both amplitude and

phase coefficients are shown for o and x disturbing polarizations. The

extension of these computations to below the gyrofrequency serves more

to illustrate the properties of the interaction expressions than to represent actual interaction experiments; the application of a ray treatment

tends to lose its meaning as low frequencies are approached, due to

partial reflection conditions and other strictly wave phenomena.

Deviative effects at lower frequencies also tend to violate certain

assumptions usually made in wave interaction experiments.

24
1

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25
10-I

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1 I I I I I

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(75)

A= FREQUENCIES USED IN CALCULATIONS

jU. 1 I 1

0.1 10 DISTURBING FREQUENCY (f D

), Mc/S

) and T for both disturbing Figure 9. The maximum values of T (+ and

polarizations as a function of disturbing frequency, showing the

heights at which the maxima occur. These calculations were

made using the Appleton-Hartree forumulas.

26
4. 4 Neglecting Self-distortion and Energy Dependences of G, v

Several calculations have been made in an attempt to show the

errors which might result from neglecting the disturbing wave selfdistortion and nonlinear absorption, and in considering G and V to be

constant at each height in (10). It is difficult to show in a concise

manner the effects of these approximations in terms of all experimental

parameters such as frequencies, power, and atmospheric profiles. In

general, it appears that these effects do not become important until

peak electron energy perturbations exceed about 100%. Under the

reference conditions (where energy perturbations do not exceed a few

percent), neglect of disturbing wave self-distortion results in only

about +0. 1% maximum error in interaction coefficient and the assumption

of constant G and v at each height results in about -. 85% error. How7 9

ever, increasing the effective disturbing power from 10 to 1 watts

increases these errors to +6. 6% and -20%, while decreasing the disturbing frequency from 10 Mc/s to 3 Mc/s yields -2. 4% and +2%.

4. 5 Dependence on Form at G(Q)

Several representations for the energy dependence of the electron

energy loss coefficient, G(Q), have been used to calculate interaction

profiles for the purpose of determining the sensitivity of interaction to

changes in G. The G(Q) functions used in these calculations are

illustrated in figure 10, and are rather arbitrary perturbations about

several dependences which have been proposed. Results are expressed

in terms of the amplitude interaction profiles in figure 11. The curve

labeled GX (Q) is taken from Megill [1965]. The curve labeled

-4 -. 55

4. 7 X 1 Q is an asymptotic approximation to Gx (Q) and appears

to give nearly identical interaction results when applied within the ionospheric electron energy range. Two other forms of G show effects on

27
ELECTRON TEMPERATURE (0), °K

232 310 387 465 542

l(T 3

10 ELECTRON ENERGY (Q), eV

Figure 10. Energy dependence of the electron energy loss coefficient, G,

within the range of electron temperatures encountered in the

ionospheric D region. Gx (Q) is taken as representing the best

available laboratory determination, and the others are various

analytical forms used in these studies. The arrows along the

ordinate indicate the constant values of G used. Electron

temperature corresponds to the rms energy given.

28
moo m ro s d c> d

O O O

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1o

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29
interaction profiles of other energy dependences. Calculations are also

-3

made for constant values of G, namely 1, 3, and 6X10 . The most

profound effects on the interaction curves apparently result from displacements of the G(Q) curves rather than changes in slope, that is,

changes in the coefficient rather than the exponent. Note particularly

the relatively small difference between the Q^Q) curve and the one

-3

using G = 3 X 10 . For the relatively small fractional changes in

electron energy encountered here, the asymptotic approximation to

Gx (Q) is satisfactory; however, for much higher disturbing powers

(or lower frequencies) the tabular representation of the Gx (Q) dependence

would be required.

4.6 Perturbations of the Electron Density Profile

Figure 12 shows the effects on the interaction profile of changes

in the electron density profile in the form of "bulges" inserted in 10 km

intervals centered at the heights indicated. The bulges represent

smooth increases in electron density of about 2^ times over the reference

profile. One of these perturbed profiles is shown in figure 5 as curve C . The corresponding interaction profiles show maximum deviations of

about 30%, which of course depend somewhat on the height of the bulge.

4. 7 Disturbing Wave Self-distortion

The degree of "self-distortion" of an originally rectangular disturbing pulse upon emergence from the absorbing region of the ionosphere is indicated in figure 13 for several variations in experimental

parameters. The initial instant of the disturbing pulse is of course

undisturbed, and the absorption increases with time, approaching the

limit corresponding to conditions of a very long disturbing pulse.

30
1

Mill km

_ km

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31
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D f D

= 3 MC/S

E ORDINARY DISTURBING WAVE

60

TIME, MICROSECONDS

Figure 13. Self -distortion of the disturbing pulse shape as it emerges from the

absorbing region of the ionosphere. At t = no self-distortion occurs,

while near the end of the pulse the distortion approaches that of the

limiting case of a very long disturbing pulse.

32
4. 8 Use of the Generalized TheorySeveral calculations have been made using the general (arbitrary

angle of propagation) refractive index expression of Sen and Wyller

[i960]. When correction factors are applied to the values of collision

frequency used in the Appleton-Hartree formulation, as indicated by

Sen and Wyller, their expression leads to nearly the same interaction

results as the complete Appleton-Hartree formulation within certain

height ranges; however, no single correction factor brings the two

into agreement over all heights. Figure 14 shows some results of

these calculations.

4. 9 Gyro-interaction Effects

Figure 9 has shown that conditions in the daytime D region do

not permit a sharp "resonance" in interaction as the disturbing frequency is varied about the gyrofrequency. However, figure 15, which

represents nighttime conditions (profile A), shows a much sharper

resonance which dominates the behavior of the interaction over a wide

frequency range. At f = f the gyrowave (extraordinary polarization) H

interaction is about 27 dB above that due to an ordinary wave at the

same frequency. Evidently, absorption in the daytime D layer does

not permit the gyrowave to reach heights where resonant effects can

become important. Inspection of the absorption expression for the

simple case of longitudinal propagation indicates that when v is at

least comparable in magnitude -with uo, the resonance in a as uo -» uuH

will not be pronounced, i. e. , it will have a low "Q":

Nv

a

v

2 + (uo- uo ) H

33
CT> OO CO LO

UU>1 '1H9I3H 9NI133IAI

I- %

z. £

LlJ - o ^

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° o

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a: s uj g

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ao

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d) >

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•* S,o

CJa

0)

cr

<uu

o

a, —

u

C r-1 M

5 rti CJ < u

(LI

34
DISTURBING FREQUENCY, Mc/s

Figure 15. Maximum values of the amplitude interaction coefficient for nighttime conditions, for comparison with figure 9 (day). Curve is for

the extraordinary disturbing polarization, and the value for the

ordinary wave at f is shown for comparison near the center of

the figure.

35
As V decreases (at greater heights), the resonant effect becomes

greater, until ray theory becomes inadequate due to effects which

cause the relation between the electron velocity distribution and the

wave phase velocity to become important.

An interesting manifestation of nighttime gyro-interaction

effects is a large increase in electron energy and collision frequency.

Figure 16 shows the maximum values of electron collision frequency

attained during the passage of a 57 microsecond gyro-disturbing puls'^,

for various values of effective vertically radiated power. Note

particularly the very high v(h) gradients at the top of each disturbed

8

region. For 10 watts, the decrease is by almost a factor of 1 in a

distance of one kilometer. Also, the disturbed region increases in

height as power increases since the decrease in the absorption coefficient at lower heights allows penetration to greater heights before

complete absorption occurs. This effect would not occur if selfinteraction were not accounted for. Daytime effects under similar

conditions would hardly be noticeable on this figure.

In spite of the greatly increased interaction at the gyrofrequency,

it is doubtful that useful ionosphere profiles can be deduced, since

almost all of the interaction takes place in an interval only a few kilometers thick. On the other hand, this property can be used to advantage

in the determination of G, V, and temperature within the interaction

region.

5. CONCLUSIONS AND LIMITATIONS

It has been shown that a numerical simulation of ionospheric

wave interaction can include several effects which are usually neglected

in analytical calculations. This technique should prove useful in the

evaluation of wave interaction experiments as well as in the development

36
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43

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CD ^»—

n rj ri a

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37
of refined theories of nonlinear ionospheric propagation. It is probable

that the present state of the theory of wave interaction is not sufficientlysophistic ate d that the results presented here represent a degree of

accuracy suitable for application to precise experimentation. The

principal virtue of this method is that it is sufficiently flexible that new

theoretical developments can be included without increasing the complexity of the basic interaction calculations. The need for such a technique has already become apparent since it does not appear

possible to make interaction calculations using even today' s most

complete theories.

Perhaps the most limiting approximation which remains to be

removed is the use of a propagation theory based on a Maxwellian form

of the isotropic part of the electron energy distribution function (the

Sen-Wyller treatment, for example). It is possible that for wave

powers used in these experiments, the distribution can become

appreciably non- Maxwellian under certain conditions. On the other

hand, collision frequencies, particularly in the lower D region, may be

sufficiently high that Maxwellian distributions are maintained even

under very high-power conditions. Layzer and Menzel [1964] have

discussed this problem for an amplitude -modulated disturbing wave,

and Budden [1965] outlines an approach to the problem, but at this

time no general solution in terms of wave propagation constants has

yet appeared.

Another problem involves the "ray theory" approximations

implicit in the present analysis; that is, that the medium exhibits only

small space and time gradients with respect to the wave parameters.

We have seen that at very low heights the heating time constant is

measured in fractions of a microsecond, a time certainly not long

compared to a wave period, particularly at lower frequencies. If

38
sufficiently high power is used that collision frequency changes at low

heights become appreciable, the ray approach may not be adequate.

Several conclusions regarding the optimization of experimental

parameters are possible in view of the present calculations. Figure 9

indicates that the optimum disturbing frequency is in the vicinity of

10 Mc/s, considering a tradeoff between available antenna gain, the

l/w interaction variation, the applicability of ray theory, the possibility

of o/x separation, and the avoidance of interpretational complexities

associated with the gyrofrequency. Figure 7 shows that disturbing

pulses shorter than about 50 [isec would not significantly improve

height resolution, and that this is really a problem of probing pulse

length. Much longer pulses are valuable in probing the higher regions

and in exploring other nonlinear effects. Variations of disturbing and

probing wave frequency and polarization can, under favorable circumstances, yield additional information, as seen from figure 8.

Figure 12 shows that the sensitivity to profile perturbations

drops off at greater heights; consequently, this experiment appears to

be most useful for obtaining profiles below about 85 km. It should thus

provide a useful extension to the capabilities of existing sweep-frequency

vertical-incidence sounders.

The utility of cosmic-noise interaction [Benson, 1962] as a supplementary experimental technique should not be overlooked. It has the capability of providing information under conditions when the

reflection technique is not useful, and eliminates the need for a smooth

probing wave reflecting layer and the inherent restriction of probing

frequencies; however, only amplitude interaction measurements are

possible, and certain accuracy limitations are imposed by the random

nature of the probing "signal".

39
An obvious extension of the numerical simulation technique is

the development of a numerical profile synthesis technique which yields

N(h) and v(h) profiles given phase and amplitude interaction data. The

two interaction coefficients are, in principle at least, capable of giving

independent electron density and collision frequency profiles. It is

easily seen, for example, that the ratio of 6a / 63 at each height

depends to a good approximation on collision frequency at that height

only. Since the pulse interaction experiment measures integrated

absorption effects between the "base" of the ionosphere and each

meeting height, any scheme for the synthesis of ionosphere profiles

from interaction data must be either a trial-and-error profile -fitting

process or an iterative procedure in which the difference between the

interaction coefficients at two heights determines the contribution due

to that height interval. Using a bootstrap approach, it is possible to

obtain the 6a and 6(3 contributions at each height from the interaction

P P

data. If the contributions due to a large number of height intervals can

be obtained, N(h) and v(h) profiles can be calculated directly. The simulation program will provide the "data" for the development of this technique.

Acknowledgement is made of the considerable contributions by

Dr. L. R. Megill to the initial development of this technique.

6. REFERENCES

Altshuler, S. (1963), Effects of inelastic collisions upon electrical

conductivity and electron heating in the lower ionosphere,

J. Geophys. Res. 6_8, No. 16, 4707-4716.

Barrington, R.E., and E. Thrane (1962a), The determination of

D-region electron densities from observations of cross

modulations, J. Atmos. Terrest. Phys. 24, 31-42.

40
Barrington, R. E. , B. Landmark, and E. Thrane (1962b), Electron

density profiles in the D-region deduced from observations of

ionospheric cress modulation, Electron Density Profiles,

NATO Conference Series 2 (MacMillan, New York) . Barrington, R. E. , E. Thrane, and B. Bjelland (1963), Diurnal and

seasonal variations in D-region electron densities derived from

observations of cross modulation, Can. J. Phys. 41, 271-285.

Benson, R. F. (1962), Cross modulation of cosmic noise, J. Geophys.

Res. 67, 2569-2572.

Benson, R. F. (1963), Cross modulation of cosmic noise: a technique

for investigating the disturbed D-region, Scientific Report

No. 1, Geophysical Institute, Univ. of Alaska.

Benson, R. F. (1964), A discussion of the theory of ionospheric crossmodulation, Radio Sci. J. Res. NBS 68D, No. 10, 1109-1122.

Budden, K. G. (1965), Effect of electron collisions on the formulas of

magneto-ionic theory, Radio Sci. J. Res. NBS 69D , No. 1,

191-211.

Fejer, J. A. (1955), The interaction of pulsed radio waves in the

ionosphere, J. Atmos. Terrest. Phys. 7 , 322-332.

Fejer, J. A. (1959)» An investigation of the ionospheric D-region,

J. Atmos. Terrest. Phys. 1_6, 291-306.

Ferraro, A. J., H. S. Lee and S. Weisbrod (1963), Phase interaction:

a new tool for D-region studies, J. Geophys. Res. 68,

1169-1171.

Huxley, L. G. H. (1953), Alternative developments of the theory of

radio wave interaction, Proc. Roy. Soc, 2 1 8

A

, 520-536.

Knecht, R. W. , and T. E. VanZandt (1964), Chapter 6 in Space

Physics, LeGalley and Rosen, Ed. (John Wiley and Sons, N. Y. ).

41
Layzer, D. , and D. H. Menzel (1965), Ionospheric cross modulation:

a microscopic theory, Radio Sci. J. Res. NBS 6_9D, No. 1,

59-68.

Megill, L. R. (1965), Self-distortion of radio signals in the D-region,

Radio Sci. J. Res. NBS 69D, No. 3, 367-373.

Phelps, A. V., and J. L. Pack (1959), Electron collision frequencies

in nitrogen and in the lower ionosphere, Phys. Rev. Letters 3,

340-342.

Rumi, G. C. , and C. G„ Little (1958), Experiment Luxembourg,

University of Alaska, Geophysical Institute Report 83.

Rumi, G. C. , and R. F. Benson (I960), Experiment Luxembourg,

University of Alaska, Geophysical Institute Report 111.

Rumi, G, C. (1961), Preliminary results of Experiment Luxembourg,

J. Atmos. Terrest. Phys. _23, 101-105.

Rumi, G. C. (1962a), Experiment Luxembourg: cross modulation at

high latitude, low height, Part I: Theoretical aspects, IRE

Trans. Ant. Prop. AP-10, 594-600.

Rumi, G. C. (1962b), Experiment Luxembourg: cross modulation at

high latitude, low height, Part II: Experimental aspects, IRE

Trans. Ant. Prop. AP-10, 601-607.

Sen, H K, , and A. A. Wyller (I960), On the generalization of the

Appleton-Hartree magnetoionic formulas, J. Geophys. Res.

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