using amplitude-modulated HF from ground an- tennas [Davis and Willis, 1974; Stubbe and Kopka,. 1977; Stubbe et al., 1981; Chang et al., 1982]. The HF...

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Excitationof the earth-ionosphere waveguideby an ELF sourcein the ionosphere V. K. Tripathi,• C. L. Chang,andK. Papadopoulos • ScienceApplications,Inc., McLean, Virginia 22102

(ReceivedAugust13, 1981;revisedMay 17, 1982;acceptedMay 17, 1982.)

It is shown that for polar regions, where the conductivity is predominantly reactive, the wave functionsfor the transverseelectromagneticmode of the earth-ionospherewaveguideare Hankel functionsof the first kind with real argument;thereforethey extendup to large heightsover the polar ionosphere. A Green'sfunctionfor the excitationof the waveguideby extendedcurrentor field sources in the ionosphereis obtainedfor an exponentialionosphere.The potential of ELF waveguideexcitation by wirelessantennasis discussed.

ally reportedextremelyencouragingexperimentalresults on ELF generationby modulating the polar There has been considerableinterestin the past in electrojetcurrent with a 4.04-MHz RF transmitter. the excitation of the earth-ionospherewaveguideat The secondtechniquerelies on resonant nonlinear ELF frequenciesby satellite-borne current sources parametricexcitationof an Alfven wave, by mixing located in the ionosphere.Galejs [1971], Einaudi and two HF waves in the ionosphere[Papadopouloset Wait [1971], Pappert [1973], and Kelly et al. [1974] al., 1982]. In both casesthe interaction region over studied this problem in considerable detail and which the ELF power is producedis comparableto demonstrated that the satellite-based antennas could the ELF wavelengthin the ionosphere,and compube radiators at least as efficientas the ground-based tations based on the reciprocity principle cannot be dipoles.One important result of theseinvestigations applied. is that the principleof reciprocityholds individually The excitation of the earth-ionospherewaveguide for all modes of the waveguideand hence the radiby extendedsourcesrequiresknowledgeof the mode ation fieldson the surfaceof the earth due to a dipole structure of the continuation of the free space transsourcein the ionospherecould be deducedfrom the verseelectromagnetic (TEM) modein the ionosphere. radiationpattern of a ground-baseddipole. Galejs[1971] hasdeterminedanalyticexpressions for 1.

INTRODUCTION

More recently, wireless ELF generation in the ionospherehas emergedas an alternative to antenna excitation.There are two main techniquesin achieving wirelessELF generation.The first relieson modulating the electrojetcurrentin the lower ionosphere, using amplitude-modulated HF from ground antennas [Davis and Willis, 1974; Stubbeand Kopka, 1977; Stubbeet al., 1981; Changet al., 1982]. The HF wavesheat the electronsand producean oscillatory electron temperature at the modulation frequency; this producesa conductivitymodulationand hencea current modulation. Stubbeet al. [1981] have actu-

i Now at Departmentof Physics andAstronomy, Universityof Maryland, CollegePark, Maryland 20742. Copyright 1982 by the AmericanGeophysicalUnion. Paper number 2S0770. 0048-6604/82/0910-0770508.00 1321

the mode structure of the TEM mode, assuming a

homogeneousionosphere.Greifingerand Greifinger [1974] examinedthe inhomogeneous ionospherecase and determined the mode structure for the case of

purely real conductivity increasing exponentially with height.While this modelgivesgood resultsfor the equatorialzones,it is of very limited validity for higherlatitudes;in actual fact, it producesvery deceptiveresults.This is causedby the fact that the ELF index of refractionis mainly real at higher latitudes,correspondingto reactiveconductivity.As it will be seenlater, this has the very profound effectof allowingthe mode to extendto much greateraltitude.This can be physicallyseenas follows:the horizontal component of the TEM propagation wave number fi is closeto co/c,which is severalordersof magnitudesmallerthan the propagationconstantI k I in the ionosphere;this implies that k is predominantly vertical.In the equatorialzones,k is perpen-

1322

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CHANG,

AND

PAPADOPOULOS

dicular to the earth's magnetic field B, and the refractive index is predominantly imaginary. At higher latitudes,B is more alignedwith k, and the ELF attenuation is substantiallyreduced. It is the purpose of this paper to study the mode structure of the TEM mode for ELF propagation at

located at a height z in the ionospherewill produce signalswhich will be attenuatedby a factor A,

higher latitudeswhere the index of refractionis pre- before reachingthe bottom of the ionosphereat a dominantly real. The conductivity is modeled as height h, whereki is the attenuationconstantgiven reactive and varying exponentially with height. The by problem is solvedanalytically,and the field structure k = k,.+ iki = nro/c can be representedby height dependentHankel funcIt may be mentioned that the concept of an attenutions. A basic difference between our solution and that of Greifinger and Greifinger [1974] is that the ation constant is strictly applicable in media with argument of the Hankel functionsis real in our case slow spatial variations.However, in media with modwhile it was imaginary in theirs, allowing thereby a erate spatial variationssuchas the lower ionosphere much higher altitude penetration. Knowledge of the it also provides a reasonableestimate of the attenuproper mode structure allows us to determine the Green's function in the presence of a 6 function source in the ionosphereand to calculate by simple integration the waveguide excitation for extended wirelessionosphericsources. The plan of the paper is as follows. In the next section we list the value of the index of refraction

as

a function of altitude for polar and equatorial regions and calculate the attenuation of an ELF wave propagating downward toward the bottom of the ionosphere. It can be seen that the attenuation is very strong at the equator, forcing us to locate any sourcesbelow 70 km, while it is weak in the polar regions, allowing excitation from much higher altitudes. Restricting ourselvesto the polar regions,we determine the structure of the waveguideTEM mode inside the ionospherefor an exponential density pro-

ationrate [Al'pertandFligel',1970].The valueof n2

is givenby 1

rt2=

2(e=cos2 0 + %• sin2 0)

{%xe=(1+ cos2 0)

+ (e•2,, + e•2y) sin2 0 _+[(e,,,,ezz(1 + cos 2 0) _ (e•2x + e•2y) sin2 0)2-4%•(e,,,, 2 2 + e,,y)cos 2 2 011/2 }

(2)

Here

2

•,,y= -- i

2

(.O p (.0c

co[(v--ico)2 + c% 2] 2

+ i

(.O pi (.O ci

CO[(Vin -- iCO) 2+

2

(.O p

(.O pi

co(co + iv)

co(co q- iVin )

file in section 3. On the basis of this structure, we

and 0 is the angle of the wave vector k with the compute the Green's function for a • function source magnetic field;top,toc, andv aretheelectronplasma, in the ionospherein section4. The last sectionsum- cyclotron,and collisionfrequenciesrespectively;vi, is marizes and discusses the relevance of our results for the ion-neutral collision frequency; and the quanwireless communications. tities with subscripti refer to ions. For the extraordinary mode (i.e., the whistler mode or the compress2. ATTENUATION OF ELF WAVES ional Alfven wave),(2) simplifiesto give the following IN THE IONOSPHERE expressionsof different heightsabove the ground. Consider the propagation of an ELF wave in the TEM mode in the earth-ionospherewaveguide.The Polar regions propagation constantfor this mode has a horizontal 2 ; ion motionnegligible c component k•-• co/c (cf. next section).In the iono- For z •< 72 kmßV2 > (.D sphere the magnitude of propagation constant k •-

n2 ,•,

103to/c;hence k• = (k2 - k•2) •/2 • k• i.e., the ELF waves propagate in the vertical direction (i.e., along the z axis). An ELF current source

For 75 • z < 95 km' ion motion unimportant

tl2,•

(-Opi

V

-coco½•cos 20 cos0+i

ELF

102

•=

SOURCE

IN

THE

IONOSPHERE

1323

To appreciate the variation of the refractive index and the attenuation factor, we plotted them as a function of z for different values of 0 (0 = 0 corresponds to poles and 0- 90ø correspondsto the equator) in Figures 1 and 2. The resultsare plotted

90 o

by usingthe generalexpression for ne and the ionosphericdata as given by Al'pert and Fligel'[1970].

Theattenuation factorexp[- • ksdz] is unityat

nr 102

z = h and decreaseswith height. In the equatorial regionsit decreasesvery rapidly with z, suggesting that the current source should not lie above 90 km in

order to have appreciable coupling with the waveguide. In polar regions the attenuation factor decreasesvery slowly with z, meaning that the current sourcecan be located at any altitude allowed by the

101

10• 60

I 80

I 100

I 120

I 140

ALTITUDE

I 160

I 180

mode structure, which will be discussednext. 200

(km)

3.

Fig. 1. Refractiveindex n, as a functionof altitude for two differentpropagationangles0 = 0ø and 90ø,at a frequencyof 100 Hz; 0 = e/2 - •bwhere•bis the geomagnetic latitude. where 2

tan 2 0 <-

V2

For 100 < z < 110km' ion motionimportantin

n2~

mvi

i v m½i

-roro½icos 20 cos0+• vi.i• (l+cos 20)

For z > 120 km

• p(• p•) n2,,•(-Dpi • [cos2 0 + 1- (sin• 0 + 4p2 CO$ 2 0)1/2]

MODAL

ANALYSIS

We model the earth-ionospherewaveguide by a perfectly conductingflat earth below z _<0, vacuum between 0 _

2rOrOci cos2 0

where 1.0

O) q- iVin pm•

O)ci

At the equator 2

n2:exx q-exy •xx

O=

0

For z •< 100 km 2

n2_•i me 0

For z > 120 km

100

150

ALTITUDE

OO pi

n2"' .•.'-7i1+ i

200

(km)

Fig. 2. The attenuation factor as a function of altitude for variousanglesof ELF wave propagation.The wave frequencyis 100 Hz, and the lower boundaryof the ionosphereis at 50 km.

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TRIPATHI,

CHANG,

AND PAPADOPOULOS

of electricand magneticvectorsfor TEM modesmay

For typical ionosphericparameters,L._• 5 km,

be written

g01/2 • 20, h -• 60 km, ro•- 102 Hz, •o = 0.4, fi, -•

as

•

•E•

Ez=(ro2•/c• -_ 112) Oz ico•/c

(3)

OEx

H•= - (o•/c• _•) Oz

(4)

whereE• is governed by thewaveequation

3z2 f12 + eE•= 0 O•E• (E•+ c• -•o•OE•,•

1.06to/c,and fii -• 2 x 10-'• km-•- i.½.,the attenuationlengthis 5000km. On changing•o •/2 to 40 and keeping other parameters fixed we obtain fi _• 1.06ro/cand fii •- 1.4 x 10-'• km-x. For a complexindexof refractionthe real part of the propagation constantis larger,and the attenuationlengthis shorter. Well-established values of attenuation of the

(5)

TEM modeare around2 dB per megameter. 4.

We take the effective dielectric constant • = ne in the

ionosphere to varywith heightas

The

• = •o exp [(z - h)/L.]

(6)

GREEN'S

excitation

of

ELF

FUNCTION

waves

in

the

earth-

ionospherewaveguide due to an extended current source in the ionosphereis best describedby a

where•o could be real or complex.For ELF waves Green's function. We consider a horizontal current sourceJ ii• in the ionosphere. The wave equation w•/e/c >>fl. In thislimit,(5)simplifies to governingEx may be Fourier transformed(or Bessel functiontransformed)in x and y to obtain

we are mainly interestedin the TEM mode for which

0z• + • •eoexp O•E•w

L.

E•= 0

(7)

and its solutionfor z • h, representing outgoing waves at z •

•, is

d2f12c21dtldExtl(o92)

dz2Ext•ro2e edz dz + '•' œ_•2 Ex• 4nifo

•x = •H{o••(0 (8)

c2 J•(z)

(12)

whereE,,t•and J• are the Fouriertransforms of E,, and J respectively. The Green'sfunction for (12) satisfiesthe followingequation'

where

• = •o exp [(z- h)/2L.

•2C21 dedG(z,Zo) H{o •) is the Hankelfunctionof firstkind of argument d2 dz 2 G(z, Zo) ro2e edz dz •, H{o ••' = dH{o •)/d•, and•o = 2(l)Ln •/2/c' Theelectric and magneticfieldsinsidethe waveguide(0 _

Hy = - iroAcos(fi'z)/cff where

if= (romic •_

The continuity conditionsat z = h yield the following dispersionrelation:

(0,2 )

4nifo

+ •- • - t•• 6(z,Zo)= - • c2

a(z - Zo)

To solve (13), we consider three regions: (I) 0 _

H, = ie•/2[B•H?)'(•) + B2H(o2)'(e)] tan ffh -

c4/•0'

(13)

(14)

(10)

These solutionsmust satisfy the continuity conat z = Zo, The right-handsideis lessthan 1, and the argument ditionsat z = h andjump conditions of the tangentfunctionis small;hence(10) simplifies to give

Hyln = Hylm +--c • _

• c

1 H[•)(•o) 2•4/: H•')'(•o)

(11)

Exln -- E•]m

(15)

Employing these conditions,we obtain the Green's

ELF

function,

SOURCE

IN

THE

IONOSPHERE

1325

Here fi and fi' correspondto the TEM mode.

G(z, Zo)= - i sin ff z

Gaussian current distribution

'fcøs fi'h[ tanfi'h cfi,•,/2 •o•(•o)lj

J,, = Jooexp(-r2/r•)•(z - Zo)

(21)

For this distribution,

4• •/'• H•oX•(•o) •o • • co•/• t'zO H•o•,(•o)• 0

ß

(16)

J•o= Joo -•'exp (-fi2rg/4)$(z-Zo) where•o - •o exp I-(•o- h)/2La] and • lies insidethe waveguide.Equation (16) has polescorrespondingto differentmodesof propagation(el. equation(10)). E•= Joo • fidfiG(z, zo)H?(fir ) exp (-fi2•/4) (22) Using (16) and taking the inverse Fourier transform in x, y of the resultingœ•agivesthe electricfield To carry out the • integration in (22), we approximate the Gaussianby a Lorentzian,

E,, =••dzo [1 G(z, Zo)J•(zo) exp JillX•_](17)

The fi integral can be evaluatedeasilyby the residue theorem. At large valuesof x•_, only the pole corresponding to the TEM mode contributes; others damp out very rapidly. Equation (17) can be expressedin terms of cylindrical polar coordinatesr, 0, z,

O.845

exp(-•r•/4)

•

(•g/4 + 0.40s)• + 0.67s

(23)

Using (23), (22) can be solved by the method of contour integration, by closingthe contour in the upper half plane. The poles at the Lorentzian occur at • • 2/% (1 ! i) and representhighly damped solutions. Consequently,the main contribution to the integral, at long distances,comesfrom the pole corresponding

E,, =.•e•nø •fo• f•o• fidfi dz oG(z, Zo)J•n(zo)H(.2'(fir) (18) •' to the TEM mode,

where

•iJoorg=

r drdoe-i•oj,, J•(fir) 1•o• •oe•

J• =•

•/•

H•(•,o) sin•'zH?(•r) c•g/• H•'(•o)

(24)

For typical ionospheric parameters in polar regions, h • 60 km, ho • 75 km, a horizontal electric

J• is the Besselfunction,J• is the currentdensityin

dipolewithIdl • 106A m andw = 102Hz produces the ionosphere, and H? is the Hankel functionof a magnetic fieldstrength H• • 10-7 A/m at a dissecondkind. For an azimuthallysymmetriccurrent tance of 1 Mm from the source.At larger distances

distribution, Jn•= 0 for n • 0; henceonlythe n = 0 thefieldstrengthgoesasr-•/2. In additionto thisthe term in (18) survives.We considernext two cases. Dipole current source

Jx = Idl$(r)$(z- Zo)/2nr

TEM mode is attenuated at a rate • 2 dB/Mm. It must be mentionedhere that the H field spectrallevel of noise at 100 Hz is • -120 dB with respectto 1

A/m(Hz)•/2[EvansandGr•fiths,1974]. 5.

In this case,

DISCUSSION

ELF waves propagate in the earth-ionosphere waveguidein the TEM mode with a phasevelocity (19) slightlylessthan the velocityof light in vacuum;the effective index of refraction of the waveguide is E,,=• fid•H•o2•(fir)G(z, Zo) tic/co• 1.2. Becauseof the continuity conditionsat the vacuum-ionosphereinterface,the horizontal comEquation (17) can be solved by contour integration, ponent of the propagationvector for ELF wavesin closingthe contour in the lower half plane and conthe ionosphere is approximately co/c.Since the efsideringonly the TEM mode at large distances, fective index of refraction kc/coof the ionosphereis Idl fi' H•oX•(•zo) • 103, the ELF wavesin the ionospherepropagate mainly in the verticaldirection.Thus we could model Idl

.l•o- • •(z- Zo)

E•-•i nx/2 ch•/2 H•oXV(•o) sin fi'zH•o2•(fir) (20)

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TRIPATHI,

CHANG,

AND PAPADOPOULOS

the ionosphere as an isotropic medium having a complex index of refraction of the extraordinary mode. The refractive index is sensitiveto the geomagneticlatitude qS;r•/2 - q5is the angle betweenk (vertical)and the earth'smagneticfield. For a current sourcelocatedinsidethe ionosphere, the electromagneticwavespenetratinginto the earth-. ionospherewaveguideare attenuatedby a factor ½xp

mentson the currentsourcemay be relaxedconsiderably. The determination of the TEM mode eigenfunctions in the inhomogeneousionosphereallows us to examine the possibilityof ELF communicationby directlyexcitingthem, usingthe ponderomotiveforce due to an HF wave, along the lines of force [Papadopouloset el., 1982]. This will be reported in a future [-I• økidz],wherehoistheheightofthesource and publication. h is the bottom of the ionosphere.Attenuationis very Acknowledgments.Discussionswith S. Ossakow and P. Palo strong for current sourceslocated in the equatorial

zone, and one is forced to locate the current sources

madessoare gratefullyacknowledged.This work was supported by NRL contractN00173-80-C-0194.

below 90 km above the ground. In the polar zones, attenuation is much weaker, and electromagnetic REFERENCES energy from current sourcesat the height of 150 km could effectivelytunnel into the waveguide.Sincethe Al'pert, Ya. L., and D. S. Fligel' (1970), Propagationof ELF and VLF War)es Near the Earth, p. 171, Consultants Bureau current sourcesare expectedto be producedby ex(Plenum),New York. ploiting some nonlinear processesin the ionosphere Chang, C. L., V. K. Tripathi, K. Ko, K. Papadopoulos,J. Fedder, and the efficiencyof the processgoesdirectly as the P. Palmadesso, and S. Ossakow(1982),ELF communicationby electrondensity,one would like to go to heightsin electrojetcurrentmodulation,submittedto Radio Sci. excessof 100 km. For this reason,polar regionsmust Davis, J. R., and J. W. Willis (1974), A quest for a controllable be used to excite nonlinear

ULV wave source,IEEE Trans. Commun.,COM-22, 578-586.

currents.

An ionospherewith an exponentialdensityprofile is found to give a mathematicallysimplemodal dispersionrelation. The eigenfunctionsare expressedin terms of Hankel

functions

of the first kind.

In the

polar zones (greater than 40ø geomagneticlatitude) the mode extendsto large heightsin the ionosphere. From the viewpoint of excitation of the earthionosphere waveguide by a current source in the ionosphere,this is an important result. This model might not give a very accurate attenuation constant of the mode as it propagatesin the horizontal direction, but that is not crucial for waveguideexcitation. The

decisive

factor

is the attenuation

of an ELF

wave as it propagatesvertically downward from the sourceto the bottom of the ionosphere.For a dipole

currentsource,an antennastrengthldl _• 106 A m seemsto be large enoughfor global communications. However, with an extended current source having directivity in the downward direction, the require-

Einaudi, F., and J. R. Wait (1971),Analysisof the excitationof the earth-ionospherewaveguideby a satellite-borneantenna, Can. J. Phys.,49, 447-457. Evans,J. E., and A. S. Grittiths(1974),Designof a Sanguinenoise processorbased upon world wide extremely low frequency (ELF) recordings,IEEE Trans. Commun.COM-22, 528-540. Galejs,J. (1971),Excitationof the terrestrialwaveguideby sources in the lower ionosphere,Radio Sci.,6(1), 41-53. Greifinger,C., and P. Greifinger(1974),Generationof ULF by a horizontalelectricdipole, Radio Sci.,9(5), 533-539. Kelly, F. J., D. J. Baker,and G. A. Chavt (1974),On the spreading of waveslaunchedby an ELF/VLF satellite,NRL Memo. Rep. 7814,Nav. Res.Lab., Washington,D.C. Papadopoulos,K., R. R. Sharma,and V. K. Tripathi (1982),Parametricexcitationof Alfv6n wavesin the ionosphere,J. Geophys. Res., 87, 1491-1494.

Pappert, R. A. (1973), Excitation of the earth-ionospherewaveguideby point dipolesat satelliteheights,RadioSci.,8, 535-545. Stubbe,P., and H. Kopka (1977), Modulation of the polar electrojet by powerfulHF waves,J. Geophys.Res.,82, 2319-2325. Stubbe,P., H. Kopka, and R. L. Dowden (1981), Generationof ELF and VLF wavesby polar electrojetmodulation: Experimental results,J. Geophys.Res.,86, 9073-9078.