Indian Journal of Radio & Space Physics Vol. 28, April 1999, pp. 87-94

Propagation of VLF electromagnetic waves penetrating the lower ionosphere

Li Kai & Pan Weiyan

State Key Lab. for Properties and Modelling of Radiowave Environment, China Research Institute of Radiowave Propagation

P.O. Box 138, Xinxiang, Henan, 453003 China

Received 31 July 1998, accepted 28 December 1998,

Propagation of VLF electromagnetic waves penetrating the lower ionosphere from above to below is analyzed. A two­dimensional Fourier transformation is employed to compute the effect on propagation of VLF electromagnetic waves by the lower ionosphere. The formula of the magnetic field on the sea surface generated by the space borne transmitter and antenna has been obtained.

1 Introduction

Electromagnetic very low frequency (VLF) waves in the ionosphere and the magnetosphere are a subject of investigations already for more than fifty years. Recently few attempts of making the experiments on the direct excitation of VLF waves in ionosphere were undertaken. The first successful experiment on generation of VLF waves with a loop magnetic antenna was conducted onboard the orbital complex "Mir-Progress-28" in 1987. It has been known l

.2 that

the programmes of the active VLF wave experiments with the uses of loop antenna (with diameter 300 m) or lipe antenna (with length of 20 Ian) on board the spacecraft or the shuttle have been presented by Russia and America.

The specific problem to be investigated is the magnetic field on the sea surface generated by currents in linear or loop antenna moving in the F2-layer of ionosphere at the height of 300-400 kIn. Several workers I.3.4 have calculated the VLF field on the sea surface generated by the space-borne linear or loop antennas. The ionospheric model in which the ionosphere is taken as a homogeneous anisotropic plasma is too simple. Taking into account wave reflections and collision attenuation in the lower ionosphere, it is necessary to calculate the reflection of the lower ionosphere from above and collision attenuation.

Pittewa/ and Tsuruda6 had used "Full-wave Method" to compute the reflection and penetration coefficients of VLF electromagnetic waves through the ionosphere. But the results cannot be used in the

calculation of the VLF fields on the sea surface generated by the space-borne transmitter and antenna.

In this paper. the ionosphere above the heights of 120 Ian is idealized as a homogeneous anisotropic plasma and the lower ionosphere below the heights of 120 Ian is regarded as a horizontally stratified homogeneous anisotropic plasma. A two-dimensional Fourier transformation is employed to analyze the propagation of VLF waves penetrating through the lower ionosphere. and the effect of collision on the ipropagation of VLF waves is also considered. Later, the ratio of VLF waves penetrating through the lower ionosphere and the VLF field on the sea . surface generated by the space-borne transmitter and antenna are calculated and computed. using the time dependence of e ilJJ

•

2 Analysis

The geometry and notation of the analysis are shown in Fig. 1; the dipole is located at the z-axis (0,0, zo). The earth' s magnetic field Bo, which has the angle OB with z direction,is in the x-z plane.

The electromagnetic properties of the lower ionosphere can be characterized by a tensor permittivity E; . The electromagnetic properties of the

ionosphere above the height of 120 Ian can be characterized by a tensor permittivity E which is similar to that of the lower ionosphere. The parameter E; is given by

... (1)

88 INDIAN J RADIO & SPACE PHYS. APRIL 1999

%

Sources l-da forn

Iono sphere c .....

~.Pt Z. =dl=1201cm

~.IIG II

II

sphere zt-l ~ .Pt

II

la-l

Ea ..... laD

Nr '1I.Pt

x.y

Seawater

Fig. I-Physical model

where, Eo is the free-space permittivity, [I] is 3 x 3

unit matrix, M i is the susceptibility matrix of the i­

layer of the ionosphere, and is given by

. .. (2)

where, U =l-iv/ro IS the effective collision

frequency of the ionosphere,

x = ro~ / ro2, ill H and roo are the gyrofrequency of

the electron and the angular plasma frequency of the ionosphere, respectively; lb and nb are the direction

cosine of the earth ' s megnetic field in the x and z direction, respectively, i.e lb =sin8B , nb =cos8 B ·

The subscripts in Eq.(l) indicates the i-layer of the lower ionosphere. In the i-layer of the lower ionosphere, the Maxwell's equations can be given as follows:

(3)

(4)

Introducing the field ' s Fourier transformations

· .. (5)

h(k,z) = 4:2 f dx ffi(r).eif .Pdy · .. (6)

--where, 1J = ~J1o/Eo is the wave impedance of free

space, the following matrix equation can be obtained.

dV. -' =ikoT ·V dz "

· . . (7)

where, ko is the wave number of free space, Vi is the

column matrix given by Vi = [exi e yi hxi hyi ]T , and Ti

is 4 x 4 matrix gi yen by

7;. 7;2 7;3 T14

T= 7;. TZl T23 T24 ,

7;. 7;2 7;3 T34 · .. (8)

~. ~2 ~3 T44

where,

LI & PAN: VLF WAVES PENETRATING LOWER IONOSPHERE 89

1:, = __ k_X_k-,-y __

24 k~ (1 + M zz )

MM 1:, =1+M _ Yl lY

32 YJ 1 + M zz

e x

e o

M M e T.=-I-M + Xl V+_Y

41 u 1+ M e zz 0

T. = -M + MxZM Zy _ kxky 42 xy I+M e

zz 0

If every layer is very thin, the layers can be regarded as a homogeneous anisotropic plasma. So, the expression of the field's transfonnation in the i­layer can be written as

.. . (9)

where, q ji and Wji stand for the eigen values and the

corresponding eigen vectors (j=1,2,3,4) of the matrix

1'; , respectively. The eigen values q ji are the roots of

the characteristic equation of the matrix Ti • The

coefficients eli' e2i and Rl i , R 2i correspond to the down-going and up-going waves, respectively. The matrix fonnulation of Eq. (9) can be written as

G(I) Ii

G (I) 2i

G(I) 3i G!: ) Cli

G(2) G(2) G(2) G!~) C 2i V;(z) = Ii 2i 3i =Gi · fi

GI~3) G(3) G(3) G (3) RJj 2i 3i 4i

G(4) Ii

G(4) 2i

d 4) 3i G!~) R2i

(10)

where, G(I) =W(I) ·eikoqli(Z-Zi-l) when J'=I,2 and I' JI

Gj? = WN) . e ikoq ji (Z-Zi) when j=3,4; WJ/} is the lth

component of the jth eigen vector in the i-layer of the

lower ionosphere and Ii is the column matrix.

· ., (II)

For the boundary at z = Z{, the following equations are obtained

V(z)1 = V ,( z)1 I z=z . 1+ z=z. , , · .. (12)

Therefore,

GI · f · =G. II · f · I l Z=Zj I 1+ Z=Z; 1+ · .. (13)

then

· .. (14)

where,

· . . (IS)

Following Eq.(l5), one may obtain

· . . (16)

where,

.. . (17)

90 INDIAN J RADIO & SPACE PHYS, APRIL 1999

When d l ~ z < do , one may obtain

(18)

where, CIO and C20 are the exciting coefficients of

the source at the height of z = do . When the source is

an electric dipole or a magnetic dipole, the coefficients may be given directly3.4.

For boundary conditions at z = d l , one may obtain

x [CI OeikOlf lf,(d, - do)

T C ikolf 20(d,-do) RIO R

20]1

20e

= [WI I W 21 WJI W 41l II =[WII W21 W JI W 4I lA 'IN · . . (19)

then

f = A - I . G - I . G . F N I 0 J 0, · . . (20)

where,

· . . (21)

· .. (22)

At z ~ d 2 , the electron density is zero and the

medium can be regarded as isotropic. In the air, the fi eld 's Fourier transformations can be expressed as follows

air B . ( ) ex = IStny z · .. (23)

· .. (24)

· . . (25)

· . . (26)

where,

y2 = kJ - k; - k;

Vai,lz=d2 = [~ PJ[ :J ... (27)

where,

~ = [sin(yd2 ) - PI sin(yd2 ) 0 P2 cos(yd2 )] T

.. . (28) and

P2 = [0 P2 sin(yd2 ) cos(yd2 ) - PI cos(yd2 )] T

... (29)

where,

k.k). 'k A I oY

PI = k 2 _ k 2 ' P2 = -k 2 _ k 2 o x 0 x

For the boundary conditions at z = d2 , one may

obtain

W]I · f =[p.. 41 z=d2

N I

· . . (30) Then

GN ·A-I ·GI-I ·Go · 10 =[~ PJ-[:J · . . (31)

Let

· .. (32)

then

... (33)

... (34)

LI & PAN: VLF WAVES PENETRATING LOWER IONOSPHERE 91

and

· .. (35)

then

· .. (36)

When the eigen values and eigen vectors of the matrix for ail layers of the ionosphere for the coefficients C IO and C20 are known, then using Eq. (36), the coefficients Bf and B2 can be easily obtained. Then, the transformations of the magnetic field on the sea surface were determined and the VLF magnetic fie"Jd on the sea surface can be solved with the inverse transform.

3 The quasi-longitudinal approximation of the field

When the horizontal wave-numbers k" and ky are givetl, the eigen values and eigen vectors of matrix T may be easily determined by the numerical method. For the VLF waves penetrating downwards from the ionosphere into free space, the allowed penetrated wave into free space is confined within a very narrow range of incident angle near the normal of the boundary, which may be called the acceptance angle for penetration6

. So, we may only consider the case which is near t.he normal of the boundary, namely,

k x ---7 0 and k , ---7 O. Then, the matrix T can be

simplified as follows.

0 0 0 -\

0 0 \ 0 T=

T11 T12 0 0 · .. (37)

T41 T42 0 0

where,

... (38)

XU(U - X) · .. (39)

. X(V 2 -12l- XV) T.41 = -1 + 2 2 b 2 2 2 . • • (40)

V (V - Y ) - X (V - nb Y )

The eigen value equation will be simplified as

· .. (41)

where,

(42)

(43)

The eigen values may be

[ [2 2]

XU(U-X)_b;

(

4 2 ) 1/ 2 Xy2 I; _ nb (~2- X)

1/2

· .. (44)

1/2

. .. (45)

... (46)

92 INDIAN J RADIO & SPACE PHYS, APRIL 1999

... (47)

The corresponding eigeJll vectors are

w,=H 2 2

IT q1 +T41 q1 +T41 · .. (48) ----

q1 T42 T42

W2 =[ - ql2

2 2 J q2 + T41 q2 +T41 · .. (49)

q2 T42 T42

W3 = [- ql,

') 2 J qj + T41 q3 +T41 · .. (50)

q3 T42 T42

W, =[ __ 1 2 2

IT q4 + T41 q4 + T41 · .. (51) ----

q4 q4 T42 T42

For the VLF waves in the ionosphere with its height above 120 km, . the following conditions are fulfilled.

x » I, y» I, X / y» I, X / i » 1, v / w« 1

· .. . (52)

When the dip angular e does not approach 1C /2,

and cos2 e· l » 1 is true, by ignoring a small

quantity, the eigen values and the corresponding eigenvectors can be simplified as

· .. (53)

· .. (54)

... (55)

... (56)

The corresponding eigen vectors are

W(2) = [-iPo - f'osign(nb) i' sign(nb) IY (58)

\ W(3) = [Po iPosign(nb) -isign(nb) 1]T (59)

W(4) = [if'o f'osign(nb) isign(nb) lY (60)

where,

Po = (/nb/y / Xt2 (61)

sign(nb) = nb / /nb / ... (62)

Here, X in Eq.(61) is a constant.

When the conditions y« viw and X« viw are

fulfilled, the eigen value equation will be simplified as

. .. (63)

Under these conditions, the wave-number of ordinary wave is the same as that of extraordinary wave, i.e., the wave would not be split into ordinary and extraordinary waves . The results indicate that the medium can be regarded as atmosphere.

One may know3.4 that the extraordinary wave, which corresponds to the eigen value qI. may efficiently go down into the air and the ordinary wave, which corresponds to the eigen value q2, is an attenuated wave. The attenuation which is caused by collision in the ionosphere may be determined by the imaginary part of the eigen vlaue q, In the ionosphere with its height above 120 km, the collision attenuation is very small; so, usually it is neglected in the calculation. In the lower ionosphere, the collision frequency is very large; so, the collision attenuation will be very large and cannot be neglected in the calculation.

From the above analysis, one may calculate the reflection and attenuation of VLF wave propagation in the lower ionosphere. Numerical calculations of the reflection coefficient in the VLF wave frequency band have been made. Figures 2 and 3 show the

)

...

LI & PAN: VLF WAVES PENETRATING LOWER IONOSPHERE 93

absolute values IR/Cd of the reflection coefficients for the lower ionosphere. From Figs. 2 and 3, it may be seen that the reflection drops down when the VLF

wave frequency or the angle BB increases. From Fig. 3, one can also notice that the attenuation of collision in the lower ionosphere is very large.

U --::. ~

13.8

13.7

0.6

13.5

13.4

10.3

13.2

13.1

\

" '. ' : '. . ...... .. , .. ... .. ( ..... ..., . ", : '> ,8~~SO ....... :- .... ... .... , ... ..... .. . ~ .. ::-.. ~ .. -!- .Io4. .. : .. :.:. j ... .

:-... : -. -.. ...... ···· ··· ·~~:..··~. ·.J!..~·d.60° .... ..... .. .... ...... " ..

. ; . . ......

.. ..... ':"'- .. ; ......... . .. , .....

~

.1. \

...•. .. . ... .. .. ... ...... .. . ....... ... .... . \ 'i

.... .. ;:" ..... ... : .. .. .... ... . : .... ......... . ....... .. . ', . :

--< ,eB~75· .. ..... -. . < .. .. ... : .~-----=-.. . -

e ~~~~~~~~~~~~~ o 5 18 15 2e 25

f,kHz

Fi g. 2-Variati on of re fl ection coefficient with VLF wave

frequency at different BR

13 .'35

0.9

0.85

U 13.8 " ~ 13.75

e.7

13.55

13.6

\ \ . \

\ \ :

" .

" " .... ... ..... \: . . .... , :

. ~,e ",45· .•.. ... . .... . . . : . . .. ...... ;. B .. ... .. . .. . ... .. ... .. . .. . ..

..... . .. ....... ..;,,; .

-~~ ~-0.55 I..I...I ..................................................................... ~ ................................ ....&..I

13 5 18 15 20 25 30 f,kHz

Fi g. 3-Variation of re tl ecti on coefficient with VLF wave at

different BH =450

[So lid line ind icates the va ri ation when co lli sion is neglected and dash line when co lli s ion is considered.]

4 The magnetic field on the sea surface

Using Eq. (36), the coefficients B) and B2 can be easily obtained. Then, the VLF magnetic field on the sea surface can be obtained easily with our method published elsewhere3

.4. We have

B2 = - 2QPoC IO . eikOq,o (d, -do)

sin(yd,)

where,

Q =_~, sin(yd)) 2Po

and

. . . (64)

. . . (65)

.. , (66)

Y2 =e-e-e · o x y , L t , being the element of the

matrix L.

In the cylindrical coordinates, the magnetic field on the surface of the sea may be obtained with an integration expression. Finally, the magnetic field on the surface may be expressed as a series. If the orientation of the magnetic dipole is along the x-axis, the magnetic field on the sea surface can be expressed as follows.

_[(2i + \)d 12[~+ i3~o - kg l~ R R R . R

J J J J

(67)

94 INDIAN J RADIO & SPACE PHYS, APRIL 1999

(68) where,

If the orientation of the magnetic dipole is vertical, the field on the sea surface can be obtained with the same method.

iM Posign(nh)

2n k5ry

[ 15 15iko ? 1 1 2( ) x -----+6k- +Ik ' R . +k I +sign(n ) R2 R. 0 0 JOb

I .I

... (69)

H I = --_. ik" A, (d, -zo) 'k f d iMQ [ d2 1 tp --0 2k e . exp I 0 q) z ,- onry d,

H (p rn)1 MPoQsign(nb ) . iA:oA,(d-zo). [ik df2 d 1 f' 'Y' z=o e exp 0 ql Z

2nry d,

... (70)

When the magnetic dipole is in general direction, the exciting coefficient CIO and the magnetic field on the sea surface are also obtained easiiy with the same method.

5 Conclusions From the previous analyses 3

.4, it is observed that extraordinary wave has small attenuation and ordinary wave is an evanescent wave. The attenuation caused by collision is very small in the ionosphere with its height above 120 km and very large in the lower ionosphere with its height below 120 km. Numerical results show that the attenuation increases and the reflection drops down when the VLF wave

frequency or the angle eB increases.

Acknowledgements Project 69671025 under which this work has been

carried out was supported by the National Natural Science Foundation of China.

References I Bannister P B et ai., AGARD Conference Proceedings 529 ,

July, 1992, 33. 1-33. 14. 2 Armand N A et af., Radiotech Electron (Russia), 33 (1988)

2225. 3 Pan Weiyan, Chinese J Space Sci.(China) , 16 (1996) 62. 4 Li Kai & Pan Weiyan, Chinese J Radio Sci (China), 13

(1998) 21. 5 Pitterway M LV et al., J Atmos & Terr Phys (UK), 28 (1966).

17. 6 Tsuruda K, J Allnos & Terr Phys (UK) , 35 (1966) 413. 7 Thomson R J & Dowden R L, J Atmos & Terr Phys (UK), 39

(1977) 879.