ac ro s s the recep t ion points. The magnitude of these changes A~(t) at sunset depends, on the one hand, on the extent of the in t e r fe rence fadings in the nighttime waveguide, and, on the other hand, it depends (just as at sunrise) on the degree of co r re la t ion between the propagation paths. Since these fac tors va ry f rom one day to the next , the phenomena descr ibed above cannot be observed on all the diurnal r e co rd s .

T h e r e f o r e , although the var ia t ions in phase di f ference of VLF radio signals both at sunr i se and at sunset a re the r e su l t of motion of the t e rmina to r along the propagation path, the nature of these var ia t ions is d i f fe r - ent. In the f i r s t ease , they a re produced by a local f ield per turbat ion in the region immediate ly adjacent to the t e rmina to r , in the second case , they a re produced by the motion of the in te r fe rence pat tern of the field in the nightt ime port ion of the waveguide behind the t e rmina to r , with success ive passages of the minima ac ross the recep t ion points. In both cases , the phase difference as a function of t ime is descr ibed quite well by the ex- p re s s ion (2), with a t ime displacement 70 de te rmined by Eq. (1).

2.

3.

4.

5.

L I T E R A T U R E C I T E D

E. A. Lewis and J . E. Rasmussen , J . Geophys. Res . , 67, No. 12, 4906 (1962). R. S. Shubova, V. F. Shul'ga, and Yu. M. Yampol ' ski i , Izv. Vyssh. Uchebn. Zaved. , Radiofiz . , 17, No. i , 43 (1974). R. S. Shubova, V. F. ~hul'ga, and Yu. M. Yampol ' ski i , in: Mater ia ls of the Second All-Union Seminar- Conference on Metrology and E lec t ron ics [in Russian] , Moscow (1971), p. 71. R. S. Shubova, V. F. Shul'ga, and Yu. M. Yampol ' sk i i , Izv. Vyssh. Uchebn. Zaved. , Radio~lektron. , 16, No. 12, 83 (1973). A'-~ ]3. Orlov and G. V. Azarnin, in: Problems in Diffraction and Wave Propagation [in Russian], No. i0,

Izd, Leningr. Gos. Univ., Leningrad (1970), p. 3.

SHORTWAVE RADIO PROPAGATION IN A HORIZONTALLY

INHOMOGENEOUS IONOSPHERE

N. D. Borisov and A. V. Gurevich UDC 621.371.25

The ro le of wave effects in propagation of shor t radio waves in a horizontal ly inhomogeneous ionosphere is cons idered . A sys tem of normal modes for the ionospheric waveguide is con- s t ruc ted with considera t ion of the effects of hor izontal inhomogeneit ies. It is shown that the expansion coeff ic ients in these modes va ry l i t t le at v e ry long path length.

The theory of radio-wave propagation in a spher ica l ly symmet r i c ionosphere can now be regarded as essent ia l ly sufficiently developed [1, 2]. However , the rea l ionosphere possesses significant a symmet ry . The cha rged-pa r t i c l e concentra t ion var ies s trongly with t ransi t ion at constant alti tude z f rom the day side to the night side, or f rom polar through t empera te to equatorial zone. Fu r the r , although such change in the ho r i - zontal d i rec t ion x occurs quite slowly as compared to the change with altitude,

= - ~ x ..~ i o - 2 , (I)

it never the less proves to be the determining fact in ve ry- long-d i s tance propagation of shortwaves. The effects of horizontal inhomogeneity in the ionosphere may also be v e r y significant in one- and two-hop propagation.

Institute of T e r r e s t r i a l Magnetism, the Ionosphere , and Radio-Wave Propagat ion, Academy of Sciences of the USSR. T rans l a t ed f rom Izvest iya Vysshikh Uchebnykh Zavedenii , Rudiofizika, Vol. 19, No. 9, pp. 1275-1284, September , 1976. Original a r t i c le submitted June 10, 1975.

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011, No part Io f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $ 7.50.

895

T h e r e f o r e , study of quest ions connected with shor twave propagat ion in a hor izontal ly inhomogeneous iono- sphere is of g r ea t in te res t [3-7].

In s tudies by one of the p r e s e n t authors and Tsed i l ina [3, 4] s imple approx imate solutions were ob- tained for the equations of geom e t r i c -op t i c s in a hor izonta l ly inhomogeneous ionosphere - the adiabat ic approximat ion . According to the approach developed in these s tudies , r a y s p ropaga te along t r a j e c to r i e s , each of which is c h a r a c t e r i z e d by i ts own value of the adiabat ic invar iant J .

The goal of the p r e s e n t study is an invest igat ion of the solution of exact wave equations in a h o r i - zontally inhomogeneous ionosphere . Section 1 will show that use of the s lowness of the r a t e of change of ionospher ic densi ty in the hor izontal d i rec t ion [(condition (1)] al lows significant s impl i f ica t ion of the bas ic s y s t e m of vec to r w a v e equations used. With cons idera t ion of Eq. (1), it is a lso natura l to expect that for cons t ruc t ion of the solution it will be poss ib le to use a s y s t e m of natura l wave functions (or no rma l modes) for a spher ica l ly s y m m e t r i c waveguide en [2, 3].* To do this it is n e c e s s a r y that the change in coeff icients of the expansion of the solution in no rm a l modes ~n during the p r o c e s s of wave propagat ion be smal l . I t develops , however , that this is not the case : as is demons t ra ted in Secs. 2 and 3, the coeff ic ients of the expansion under r e a l ionospher ic conditions change grea t ly a t d is tances of ~ 1 0 2 km. This means that it is difficult to use the s y s t e m of normal modes of a spher ica l waveguide Cn for de terminat ion of shor twave f ields not only for v e r y longpaths , but even for one-hop paths .

A no rma l mode s y s t e m adequate for the p rob lem cons ide red mus t cons ider horizontal inhomogeneity of the ionosphere f r o m the v e r y outset . Such a s y s t e m k~N in the local ly homogeneous (adiabatic) approx i - mat ion is cons t ruc ted in general f o rm in Sec. 4. It will be shown that for long path, and even for antipodal propagat ion , the coeff ic ients of the expansion in ~N change only slightly. I t is thus natura l to t e r m the mode s y s t e m ~ adiabat ic .

Compar i son of the r e s u l t s obtained with the adiabat ic approximat ion in geome t r i c -op t i c s (See. 5) shows that in wave packets sufficiently s p r ead over J the t heo rem of conserva t ion of the adiabat ic invar iant is fulfilled sa t i s fac to r i ly on the whole.

1 . B a s i c E q u a t i o n s

The propagat ion of e lec t romagne t ic waves is desc r ibed by vec tor wave equations. If the d ie lec t r ic pe rmi t t iv i ty a of the p l a s m a is dependent on only one of the coord ina tes , in the solution of the p rob lem the f ield may be broken down into vec to r components . The bas ic equation then r educes to independent s ca l a r equations for individual components (see, for example , [1, 8]). In the general case where ~ is a function of two or th ree coord ina tes , it is not poss ib le to obtain a s y s t e m of independent s ca l a r equations. How- e v e r , given the condition that the d ie lec t r ic pe rmi t t iv i ty of the ionospheric p l a s m a changes sufficiently slowly in the hor izonta l d i rec t ion [see Eq. (1)], it is convenient, as before , to seek the solution by s e p a -

r a t ing the f ield into vec t o r components .

Thus , we will l e t the d ie lec t r ic pe rmi t t iv i ty e be an a r b i t r a r y function of coordinates e = e (r). F r o m Maxwel l ' s equations for a space f ree of sources we have the express ion for magnet ic field H:

rot 1 rot H - - K0 2 H = 0, (2)

where K 0 = w / c ; w is the wave frequency; and e is the veloci ty of light. Substituting in Eq. (2) for H the f o r m H ~ r o t A 1 , a f te r s eve ra l t r an s fo rm a t i ons we a r r i v e at the equation

~A,-- (V In~) divAt + ~K'~A, = ~v r (3)

Here ~ is an arbitrary scalar function which appears after application of the operator rot in Eq. (2). In the special case of spherical symmetry $ = e (r), setting A i = erA1, $ = 2Al/er , we obtain the well-known

equation [I , 2]

d 1 3 1 sinO A~+aK~A~=O. (4) .Or ~ Or + r ~ stn-------~

This equation desc r ibe s the field of a wave exci ted by a rad ia l cu r ren t .

A second independent solution may be sought by introducing the potential A 2 such that the e lec t r i c f ield E is e x p r e s s e d in t e r m s of A 2 as follows: E ~2) = ( 1 / e ) r o t eA z. Substituting E (A2) in the equation

*Such an approach was used recent ly in [11].

896

we find for the potential A 2

rot rot E ~ * K~ E = 0,

AA.,--rotlx' ln*, A2] + s ~ o 2 = r r

(5)

(6)

where ~ is again an a r b i t r a r y sca la r function. In the case of spherical symmet ry , introducing A 2 = erA2, = 2A~/r , we have

I de 1 a 00_ 1 00_~,2 ~ 0-~-2 + r2sin-------- ~ 00sin 0 o + r ~ s i n 2 0 A.z--~K~A.,=O. (7)

Equation (7) defines the field of a wave excited by a vor tex cur ren t lying in the plane O, ~o.

For s implici ty, we will now l imit the examination to the plane problem. Considerat ion of spherici ty, as pe r fo rmed in Sec. 5, does not lead to qualitatively new resul ts . We assume that the dielectr ic pe rmi t - tivity e = a (z, x), fur ther assuming that in the horizontal direct ion x the dielectr ic permit t iv i ty changes sufficiently slowly, and that at any x it has a maximum with r e spec t to the ver t ical z, so that a wavegx~ide channel in which a wave can be maintained is formed.

form

The system of equations for the components of the vector potential which follows from Eq. (3) has the

Oz ~ + Ox ~ Oz t Oz + Ox /

O"-A, ( OA, Oz"- + ax ax \--g2-z + -Ex / + = o.

(8)

In the case where s is not dependent on horizontal coordinate x, Ax = 0 and sys tem (8) reduces to one equation for Az. In a horizontal ly inhomogeneous medium, consider ing Eq. (1), we can seek a solution

of Eq. (8) in the form of a series in powers of fl:

_= A In)

In the lowest approximation in fl we have the following equation for the basic component of the potential Az:

0 1 OA~ @A~ . 2

"Oz .~ Oz + ~ + ~ O. (9)

The solution of Eq. (9) will be const ructed in the following section.

2 . E x p a n s i o n in N o r m a l M o d e s o f a H o m o g e n e o u s W a v e g u i d e

We introduce the new potential A such that Az = ~--aA. Then for A we obtain the equation

02A o OA 0in:- (10) 02A + + K;UA OP ~ Oz Ox

where U = e - [(3/4e2K~)(0 s / 0 z ) 2 - ( 1 / 2 a K ~ ) ( 0 a s / 0 z 2 ) ] . For shortwaves we may consider U ~ a (see (31). We tempera r i l y re ta in in the r ight side of Eq. (10) the small t e rm of order fl ; it will be shown below that

this t e r m may be neglected.

We consider a sys tem of or thonormal ized functions ~0n which are eigenfunctions of the homogeneous

problem corresponding to bound states for the equation

0~%.. + (K~o~ -- k,2,) %, = O. (11)

The functions qan in the horizontally homogeneous (or spherically symmetric) case ~ = e (z) form a com- plete system of natural modes of the ionospheric waveguide [2-4]. In this case the solution of Eq. (i0) is represented in the form of superpositions of normal modes A = ~'n Cnq~ with constant coefficients Cn.

In a horizontally inhomogeneous waveguide the coefficients Cn are no longer constant. Then the solu-

tion of Eq. (ii) may be sought in the form

A = ~ c,,(x)~nexp ( - - i fk,~dx) 1 -. (12)

897

The eigenfunetions q~a depend on x and on the p a r a m e t e r and for any fixed x sa t i s fy the re la t ionship

( ~'~'~ "~ ~ S dz ,% ~ = .~,,~. (13)

We note that the solutions ~n, s ince they co r r e spond to local ized s t a tes , may be cons idered r ea l . A s s u m - ing that the eoeff ieients c n do not change too rapidly with x, I d l n e n / d x l << K 0, with considera t ion of Eqs. (12), (13) we a r r i v e at an equation for en:

dc, . . . . Ha'" exp [i S (le.-- k,,,)dx] c.~, (14) dx

~n

where

. . . .

"" 2 \ ' " ' ' o x /

Here the b racke t s ( ) denote averag ing over z, as in Eq. (13). Equation (14) defines the evolution of the initial s ta te upon wave movemen t through the channel.

In the case where e(z , x) is c lose to unity, as is usual ly t rue in ionospheric wave channels , the c o - eff ic ients Hnm (1) and Hnm (2) a re not independent, but a r e in te r re la ted . In fact , we different ia te Eq. (11) with r e s p e c t to x, mu l t ip lyby Cm, and in tegra te ove r z. Then we mult iply the equation for r by 8~n/OX and again in tegra te ove r z. Compar ing the two r e su l t s , we find the des i red re la t ionship:

H,~, 1 / d~ , = 1 [ Ok~/Ox k'~ k: 1 nm -'~ - - \ ~ . . ~ . , - - / ..... ~nrn -+ ' " " - ' C ~'~,~ . ( 1 5 ) 2 , K0

Z~

We note that in the quas ic lass ioa l approx imat ion for each of the no rma l modes Cn the quantity I (K2~ - Zl

k2)l~dz (z 1 and z 2 a r e the turning points) is constant"

p r ~ "

J P K,,-: -- k2-dz, = = n + . (16) : t

At l a rge va lues of n this constant coincides within a numer i ca I mui t ip l ie r with the adiabat ic invar tan t J

int roduced in [3, 41.

Wave propagat ion in the channel ean be eonveniently descr ibed in t e r m s of no rma l modes ~o n only in the ease where the expansion coeff ic ients en change stffficiently slowly along the t r a j ec to ry . Thus to con- tinue fu r the r it is n e e e s s a r y to de te rmine the values of Hnm0) , Hnm (2) for a r e a l ionosphere .

3 . E s t i m a t i o n o f M a t r i x E l e m e n t s H n m f o r a R e a l I o n o s p h e r e

Short radio waves with f requencies s ev e ra l t imes g r ea t e r than the p l a s m a frequency and the e lec t ron gyrofrequency can propagate in ionospheric wave channels [3]. The d ie lec t r ic pe rmi t t iv i ty for such waves d i f fers l i t t le f r o m unity, ~ = 1 + 6 e , t s ~ I < 10-1, so that the depth of the channels which develop is quite smal l . Neve r the l e s s , the number of na tura l modes is usual ly quite l a rge : N ~ 102-10 ~. The alt i tude range over which va r ious ionospheric channels exis t c o m p r i s e s 100-350 km.

To e s t ima te the m a t r i x e lements Hnm we will cons ider a region in the vicini ty of the channel min i - mum - the launching point z = ZL. He re the d ie lec t r ic pe rmi t t iv i ty e may be cons idered as behaving parabol ica l ly [3]: ~ = ~ M - 1 / 2 ~ / ~ ( Z - Z L )2" We t r a n s f o r m to d imens ion less va r i ab l e s ~ = ( Z - Z L ) / p L, En =

~K0et ~ .~-lv~ kn,, _2 ~n2L, ~n L- -- L[2/a ~vl ~"~K2.]1/4- Equation (11) in these va r i ab l e s appea r s as

d2~'" .-t- ( E n - .:2) ~ . = 0 (17 ) d ~2

and has solutions e x p r e s s e d in t e r m s of H e r m i t i a n po lynomia ls

%, = s,, e-~,/2 H,(~), (18)

where s n = (~-'~n~.2n) -1/2.

Consider ing that E n = 2n + 1, we p r e s e n t an expres s ion for wave numbers kn:

_ _ K . "~ (2n + 1). g9}

898

The di f ference between two consecut ive wave number s is _" 112

t , k = l e ~ _ j - - k,, = (2-~-M) , % ~ (20)

With the aid of Eq. (18) we can wr i te the quantity 0~Om/~x appear ing in Hnm (1) in the f o r m

c) ?___~ _ e -~'- mH,,,-l'-- ~ H,~,-i a(x), (21) t~X - - Sm

where a(x) = - ( ! / p L ) (dzL/dX). Using Eq. (21) and the definition of Hnm(1), we obtain

rg, j , . rn ~- 1 ~ ... . . +1 - (22) , , r im ~ a (X ~n, r n - - I - 2

Here 6nm is the Kronecker delta, and n, m are the numbers of the modes excited.

We will estimate the value of a(x) for an interlayer ionospheric channel. A peculiarity of this channel is the fact that on the night side it rises much higher above the earth than on the day side, so that the alti-

tude change ZLmax-ZLminreaehes I02 km [9]. For the estimate we take ZL(X ) = ZL (~ + 5ZLCosqox, where ZL (~ is the mean altitude above the earth, and 6ZLiS the deviation from the mean value, qo = I/Ro, and R 0

is radius of the earth.

The expression for a(x) then takes on the form

a(X) ~ q ~ 1 7 6 s i n q a x . (23) 2L

We will p r e s en t r e s u l t s of the a(x) computat ion for two concre te c a se s . Using the ionospheric model cons ide red in [3], for a f requency of ~ = 1.5 �9 108 sec "t we find a(x) = 10 -2 sin q0 x (km-1). With the aid of expe r imen ta l e lec t ron concent ra t ion p rof i l es at the equator , taken f r o m [10], for a f requency of o~ = 1.25 �9 10 8 s e e - I we find a(x) = 3 - 1 0 -3 sinq0x (kin-i).

Thus , for modes sufficiently high (m >> 1) we obtain f rom Eq. (22) a quite l a rge per turba t ion [Hnm(i)[ ~ !0-z-10 -1 km -i . To in te rp re t this r e su l t we rewr i t e Eq (15) in the fo rm

H',,'~z- Ko 1 ( ~ n -- m. (24) k ~ k,, 2 ~'~" O x / '

I t follows f r o m Eq. (24) that the per tu rba t ion Hnm (1) is l a rge for adjacent modes [ m - n J = 1 in the shortwave reg ion . Also significant is the p r e s ence of l a rge changes in the alt i tude at which the in te r layer channel l ies . With inc rease in the di f ference [ m - n [ the m a t r i x e lements Hnm (i) decay rapidly . In fact , if we use the quas i c l a s s i ca l exp re s s ions for r and q~m it is evident that with growth in I m - n [ the integrand in Eq. (24) begins to osci l la te rapidly . Moreove r , the fac tor [ K 0 / ( k n - k m ) i d e c r e a s e s s imul taneously .

Hnm (2) From Eq. (24) it also follows that for m ~n, ~ [(kn-km)/K0]Hnm(1). According to Eq. (20) for adjacent modes Ak ~ (e~/2)I/2 which for ionospheric channels means Ak ~ (2-3). 10 -3 km -1. In the

shortwave region the quantity Hnm (2) is of the order Hnm (2) ~ 10-SHnm (1), i.e., negligibly small. At m = n we may assume Hnm (2) = I/2 < q~n(0 In ~ / ax) ~ i0 -~ km -i, which is also too low. Therefore, in the

future we will neglect the quantity Hnm (2) in comparison to Hnm [ ~.

Thus, for shortwaves the interaction of Hnm between adjacent normal modes in the real horizontally inhomogeneous ionosphere proves to be quite strong. The coefficients e n change rapidly with wave motion,

IHnml the modes become confused~ and perturbation theory ceases to operate at distances of the order of -I 10 -2 kin, i.e., significantly less than one hop. Thus, in fact, the normal modes Cn cannot be used for

calculation of the wave field for shortwaves.

With increase in wavelength the interaction between modes becomes weaker, since according to Eq. (22), (23), [Hnm(1)[ ~ kl0/2. Consequently, for radio wavelengths the situation differs, and it is quite con-

venient to use the modes q~n in studying radio-wave propagation.

4. Adiabatic Mode System of a Horizontally

Inhomogeneous Ionosphere

The results of the preceeding section indicate that for shortwaves the natural mode system must be constructed with immediate consideration of the effect of horizontal inhomogeneity, i.e., with consideration

899

of the Hnm interact ion. On the other hand, since the p roper t i e s of the ionosphere change slowly in the horizontal d i rect ion [Eq. (1)], it can be rega rded as local ly homogeneous at eve ry point.

With considerat ion of J21is we cons t ruc t a new sys tem of natural modes in a horizontal ly inhomogene- ous ionosphere, which it is natural to t e r m adiabatic. We will consider the function ~ = ~ anfN)q~n, the

coeff icients a(nN) and corresponding eigenvalues KN of which a re determined f rom the chain of equations

(h~ K) a , i ~9. u c1~ ~, = O, ( 2 5 ) ~ ~ * * t ~ r t l t ~ r l t

m

x (*) Each value of KNis a r o o t o f E q . r We note that the ma t r i i i tnm is t Iermit ian. The number of such func- tions ~o n which produce a significant contribution to ~N will be denoted by An, where An ~ IHn,n_I/AKI. With increase in the dif ference I K N - k m l fo r m such that IHm,m_l /KN-kml < 1, the coefficierlts am d ee rea se r ap - idly. In o ther words, new modes a re quite broad packets of old modes ~o n.

The eigenvalues K N and coefficients an, as follows f rom Eq. (25), a re periodic functions of x:

a~(x) = a,(x-2r. Ro), g.v(x) = Ktc(x + 2r. Ro). (26)

The new eigenfunetions belonging to the var ious eigenvalues K N are orthogonal at any fixed x:

<tI'~ ~'~r) = ~te~f" (27)

We wri te the general solution of Eq. (10) in the following form:

t F = ~ , w ( x ) ~ ' v e x p ( - - i K v d x l ~ . (28)

Substituting Eq. (28) in Eq. (10) and using Eq. (27), we find the equation for the coefficients aN:

"-da:-d~N + E~ i/FK[)i-~hv,,(xjex p [-- i ((K,~ - KN)dx I ~ , =-0, (29) t /<,~ '

. it

where

t~ ~X

In deriving Eq. (29) it was considered that k2n-K.~q ~ 2KN(kn-KN).

If by using Eq. (25) we evaluate the characteristic harmonics of the Fourier series for the coeffi-

cients [a(nN) = ~ ~ l cos/q0x] which produce the basic contribution to 9N, it can be shown that the quantity

l has a value of several units. The hNM interaction is of the order of IhNMI N i/Ro. There now appears a new small parameter T -- IhNM/zkKI << I, which allows use of perturbation theory in solving E q. (29). For example, let the initial state be given by the condition aM(0) = 6NM. Then for x > 0 it follows from

Eq. (29) that

"Jaf --O IN- -MI~K ' ~.v AK ]"

Thus the hNM i~teract ion does not lead to significant changes in the initial state local ized over modes tIN.

We now turn to the second equation of (8). We set A x = ~ ~N(X)q*Ne-ifKNdx(1/~KN ). T h e n f o r d e t e r -

m i ~ t i o n of ~N we have the inhomogeneous sys tem of equations

Ill

M

with initial conditions qN(0) = 0 for all N. Since the coefficient [(1/2KN)(a l n ~ / 0 x ) i < 10-7 is ex t remely smal l , in the p resen t study we will not consider the potential component A x.

9 0 0

5. Radio-Wave Propagation with Consideration of

the Earth's Sphericity

We will turn briefly to a description of the wave field in spherical coordinates. We direct the polar axis perpendicular to the plane of propagation. Let the wave be excited by a radial current localized near the earth. The electromagnetic field of such a wave will be described by Eq. (3), while in the ionosphere where 8r ~ 0 it will be necessary to introduce two components of the vector potential: A = {A r, 0, A~}. However, considering the smallness of the parameter fl = I (I/r)(8 r / 8 r r I, we can, as before, reduce the problem to solution of one scalar equation which coincides with Eq. (4) at 8Ai/80 = 0. Intro- ducing the new variable A (see [3]) such that A I = v~r (R 0 + z)A, we obtain the approximate equation

r)2A 1 c)~A , 2 +~ K0A =0, (32)

where g' = r [i + (2z/R0)]; z is the altitude above the surface of the earth. The only difference from the preceding ease is that in Eq. (32) there appears the effective dielectric permittivity g', produced by the presence of curvature. The analysis of the solutions of Eq. (32) is no different than that performed above.

6. Comparison with the Adiabatic Approximation of

Geometric-Optics

In conclusion, we will compare the results obtained herein with the adiabatic approximation in geo- metric-optics [3, 4]. As was shown in Sees. 2 and 3, the coefficients of the expansion of the solution over normal modes of a spherically symmetric waveguide in a horizontally inhomogeneous ionosphere are not preserved. Effective mixing of adjacent modes occurs. However, since according to See. 2 the number of modes is equivalent to the adiabatic invariant J, this means that due to wave effects, wave energy re-

distribution over adiabatic invariants takes place.

How significant is this redistribution? This question may be answered by analysis of the system of adiabatic normal modes constructed in Sec. 4. The functions ~N are packets of modes Cn having an effec- tive width over n equal to An ~ [Hn,n_I/~K I. Under ionospheric conditions, &n ~ (1-3) �9 10 I. The expan- sion in functions of ~N is preserved on long paths and even antipodal paths. This means that wave packets over n having a width zkn are preserved, or, in other words, packets over the invariant J, having a width

~j ~ (~n/n)J.

Thus due to wave effects the wave energy is redistributed over adiabatic invariants to a width of the order of AJ. We consider, however, that the total number of wave modes in the ionospheric channel is very large (nmax ~ 103). Therefore, An/n ~ 10 -2 and &J/J0 ~ 10-2, where J0 is the maximum adiabatic invariant of the channel. This means that for a sufficiently smooth distribution over invariants of the wave energy in the channel, its redistribution due to wave effects is practically insignificant. In other words, the geometric theorem of preservation of adiabatic invariant under the given conditions is fulfilled satis-

factorily.

LITERATURE CITED

i. H. Bremmer, Terrestrial Radio Waves, Elsevier Publishing Company, Amsterdam (1949). 2. P. E o Krasnushkin, The Normal Mode Method and Its Application to Problems of Long-Distance

Radio Circuits [in Russian], MGU, Moscow (1947). 3. A.V. Gurevich, Geomagn. A6ron., ii, 961 (1971). 4o A.V. Gurevich and E. E. Tsedilina,--Geomagn. A6ron., 13, 283 (1973). 5, T. S. Kerblai and E. M. Kovalevskaya, Geomagn. A6ron.~-7, 123 (1967). 6. A.G. Slflionskii, Preprint No. 12, Inst. Zemnogo Magnetizma, loaosfery, i Rasprostran. Radio.,

Akad. Nauk SSSR (1971). 7. V.I . Sazhin, Yu. A. Semenei, and M. V. Tinin, Sb. Issled. Geomagnetisma, Agron., i Fiz.

Solntsa, No. 32, 53 (1974). 8. J.A. Stratton, Electromagnetic Theory, McGraw-Hill (1941). 9. i.A. Tushentsov, D. I. Fishchuk, and E. E. Tsedilina, Geomagn. A6ron., 15, 78 (1975).

i0. T.N. Soboleva, Preprint No. 16, Inst. Zerrmogo Magnetizma, Ionosfery, i Ra---sprostran. Radio.,

Akad. Nauk SSSR (1973). II. A.A. Malyukov, I. I. Orlov, and V. N. Popov, Geomagn. A~ron., 1__5, 370 (1975).

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