Cavity resonances for a spherical earth with a concentric anisotropic shell

Journalof Atmospheric andTerrestrial Physics, 1965,Vol.Zi,pp.81to89.Pergsmon Press Ltd.Printed inNorthern Ireland

Cavity resonances for a spherical earth with a concentric anisotropic shell JAMES R. WAIT Central Radio Propagation Laboratory, National

Bureau of Standards, (Received

28 July

Boulder,

Colorado

1964)

Abstract-A boundary-value problem of a concentric spherical system involving anisotropic media is considered. Rather gross simplifications are made in order to achieve tractability and yield physical insight. The principal idealization is the representation of the concentric The derived results have some relevance to the plasma layer by a thin shell of ionization. phenomenon of resonance in the concentric cavity formed by the earth and the ionosphere. For example, it is shown that the earth’s magnet,ic field and the electron collisions both contribute to the damping of the cavity resonances.

1. INTRODUCTION PROPAGATION in anisotropic media is a subject which is fraught with analytical difficulties. General theoretical treatments which make no approximations usually lead to intractable results. While much progress has been made by the use of highspeed computers there is still a need to study idealized situations (e.g. WAIT, 1961a and 1963) which provide physical insight. A problem which falls in this category is the determination of the resonances in the region between the earth and the lower edge of the ionosphere. A full or complete solution would require that both vertical and horizontal electron density gradients be considered. Furthermore, the variable strength and direction of the earth’s magnetic field would need to be considered. In most studies (GALEJS, 1963; THOMPSON, 1963; WAIT, 1964) to date, the horizontal electron density gradients and the variability of the magnetic field are consistently neglected. This idealization is also contained throughout the eminent treatise by BUDDEN (1961) which is confined to waves in inhomogeneous anisotropic plasma, but with a constant d.c. magnetic field. An ingenious analysis which allows the d.c. magnetic field of the plasma to be non-uniform has been given by HOFFMAN (1964). His results, after making WKB-type approximations, are applied to the interpretation of observed geomagnetic fluctuations in the magnetosphere of the earth. In the present work, a relatively simple model is employed to discuss certain features of the earth-ionosphere resonance. In particular, the influence of earth’s magnetic field is displayed in a clear manner. Some suggestions are also made about the possibility of treating the same problem when the horizontal gradients are included in the analysis. 2. FORMULATION

The model adopted as indicated in Fig. 1 is a perfectly conducting sphere of radius a, surrounded by a concentric anisotropic shell of radius b, and thickness d. In terms of spherical coordinates (r, 0, c$), the inner spherical surface is r = a, and 6

81

r

The properties of the thin shell are described M and N, which are defined by

in terms of two dimensionless

paramekess

(1) where Y is 01~is tr), is Q is

the the the the

collision frequency, (angular) plasma frequency, radial component of the gyrofrequency, __ dieleotric constant of free space ( == 8*85-t x 1O--iz).

and ^/Iis the characteristic

impedance

of free space (g. 120 n-).

When the d.c. magnetos field is removed 01, vanishes and as a result N may be written in the form M = (a&r], while .!Y becomes zero. IHere, O, = +w~~/(Y +- iw) is the equivalent complex conductivity of t,hc isotropic shell of ionization. Thus, Q,&is the admittance of the shell which is convertient normalized by multiplying by 71. E’or convenience of discussion, the region T :‘- h is described by ;2 subscript 1 on various field components, while the region CL-c: 7 < b is described by a subscript; 2. With this notation the jump conditions for the magnetic fields at the sheet are ---HE, NE’,

-- SE,

= ?#i, $4--- H,,)

(3

~- ,VF, = ?](H,,, ‘-.’ FI,,J I T_li ?

where E. and E, are the electric fields within the sheet.

P)

Because of the assumed

Cavity resonances for a spherical earth with a concentric anisotropic shell

thinness of the shell, the tangential electric fields are continuous. Thus,

E,=El4 =E,, 1r+, E. = El0 = Ego

As a result of the assumed perfect conductivity are E, = 0

83

(4) (51 (6) (7)

of the core, two additional conditions

E,=01r’=a *

3. METHODOF GENERALSOLUTION

To seek a solution in the general case, it is conventional to express the fields in region (I) and region (2)in terms of spherical wave functions. Recognizing that these fields satisfy Maxwell’s equations, they may be derived from two scalar functions, U and V, which, themselves, satisfy (V2 + k‘q :j = 0,

(8)

where k = .%/free-space wavelength. For example, the radial field components are related to U and V by E, = (kz + $)(rU), (9) and $7, = (k2 + ~~)~~V~.

(10)

A general representation for U and V is obtained by superimposing elementary solutions of the type ~~(k~} P,na(eos 8) exp (~~~), where z, is a spherical Bessel function of order n and P,” is an associated Legendre polynomial of order n and degree m. In the general case the summations extend over all integral values of m and n. The unknown coefficients are to be de~rmined from the boundary conditions as stated above. The problem is greatly complicated when M and N are functions of 8 or 4. For example, if the d.c. magnetic field is produced by a,magnetic dipole centered at the origin, CU,would vary as cos 8. The 0 dependence of M and N then cause coupling between the various modes of order n. Furthermore, if the electromagnetic fields are excited by a dipole which is non-aligned with the axis of the magnetic dipole, the fields will also be dependent on $. This results in further coupling between the various modes of order m and n. 4. TWE AZIMUTHAUYSYMMETRIC PROBLEM

In order to simplify the whole problem, it is assumed that the parameters of the plasma sheet do not vary with 0 and #. Thus, J4 and N are regarded as eons~nts. Then, without further loss of generality, all field quantities are taken to be independent of 6, Strictly speaking this would correspond to excitation by a radially oriented dipole at 8 = 0. However, more general forms of excitation may be considered by superimposing such solutions.

ii,l,fll and h,,(“i are spherical Hank4 functions of the first and second kind as conventionally defined {e.g. WAIT, 1962). The coefficients A,,. B,, 17?(,G,~.6,, and c,, are to be determined from the boundary conditions. The radial field components are obtained from (9) and (lo), while the tangential field components are obtained from Here,

(15)

Utilizing these equations, the boundary conditions, written explicitly in terms of CTand V a8 follows

as defined by (2) to (T), may bc

(17)

(18)

(19) 5. THE EXPLICIT SOLCT~ON In what follows, it proves convenient to introduce a more compact notation for the spherical wave functions and related parameters. Thus, the functions u and v are defined by u(kr) = (~~) ~,~(I)(~~}~ (20) v(rcr) =I (Lr) hR‘2’(kr), and the pertinent values of Icr are denoted by x = ka and y = kb.

(21)

Cavity resonances for a spherical earth with a concentric anisotropic shell

85

The boundary conditions denoted by the pair (19), when applied to (13) and (14), immediately lead to the relations B,

=

$

-

A, and b, = - g

a,,

(22)

where the prime indicates a derivative with respect to x. Using (22), it is a relatively straightforward matter to show that (16) to (18) lead to four simultaneous equations in the coefficients. This set may be conveniently written as follows:

Qj1-A. + QjgCn + Qjga, + QitCn = 0, where j = 1, 2, 3 and 4. Explicit

expressions for the Q’s are :

&I2 = -Mu’

and, finally,

(23)

(25)

+ iv(y),

Q,,= --iN,@A,

(26)

Qzz= --NV’(Y)>

(27)

-

$$V'(Y),

Qzs=

U'(Y)

924=

~Mv(Y)

-

V'(Y)>

&a =

-U'(Y)

+

$

Qsz=

V'(Y)>

Qm =

-U(Y)

Q,,=

V(Y),

(29) (30)

V'(Y),

(31)

+

E

(32)

V(Y),

(33)

Q1, = &,I= Q33= &a‘, =

&a = Qa = 0.

(34)

In order that a non-trivial solution exists for the coefficients A,, C,, a, and c,, it is necessary that the four-by-four determinant of Qja vanishes. When the index n is an integer, such a determinantal equation may be solved for the complex frequencies Q which, by definition, characterize the cavity resonances of the system. An explicit form for this cavity resonance equation is NY)

-

u’(x) ~

v’(x)

fJ(Y)

X ([u'(Y)

1

-

- g

c U’(Y)

~

V'(Y)

aY)]+ g - $$wY~ 1 + U'(Y V’(Y)

- V'(Y)l)

u’(x) U(Y) ~-~

u(x)

v’(x) I[ V(Y)

V(X)1

N2v(Y) v’(y) = 0.

(35)

6. LlhlI’l OF I~X’I’KEJIEI,~- Lo\\

FfLEQI~ESCI

fcs

In the present paper tlke modal equation xvii1 IX considered in the limiting Case of extremely low frequency such that the difierence h :m ~1 .I’ is small compared with unity. In this casC, the functions u(!Y) and I may be convcnientjlv esprcssed as a Taylor series in t,hc manncl U(Y)

U(Z) -;-- 6?L’(Z) f (h~i’)ec”(x)

/

.. ,

with a similar expansion for v(Y), u’(Y) and v’(Y). Some simplification noting that u(x) and V(X) sat,isfy t’he cquatioli M”(Z) -I E,U(.z.) :

is achieved

I).

by

(36)

where x,, :-x while U(X) and V(X) satisfy

.r2

a Wronskian U’(X) w(x) -

Then, on neglecting

Using these low-frequency considerably to

modal

V’(Y) ----- = d ki

V(Y)

relation

)

(3’;)

_ 1 of the form

(38)

U(X) V’(X) := 2i.

all higher order terms, it, is found that td’(Y) o’(z) -

This form of the defined bv

‘)

1 _ 57

i

(39)

u’(x) e’(j/) = SST,,,

U(Y) O(X) - u(x) v(y) ;= L’is.

($0)

u’(y) v(x) -- u(x) v’(y) ^_ “i.

(41)

u(y) v’(z) -

(42)

u’(z) v(y) ‘2 -3.

approximations,

equation

involves

log v(Y) = iG log

the

modal

equation

the

logarithmic

(35)

simplifies

derivative

function

(n + WL)! e-iy $ 1,!,_om!(n - m)!(2iy)” [

1.

Cavity resonances for a spherical earth with a concentric anisotropic shell

Two specific cases are

_ 1 =_-i V’(Y)

[ V(Y)

(45)

n=l

and 3

L-1 V’(Y) V(Y)

87

n=2

= -_z

6

6

l + iy + -(iy)2 + -(iy)3 3 3 1+iY+(iy)Z

1 *

(46)

It may be seen that, in general, the ratio v’(y)/v(y) approaches -i for y > n. Unfortunately, in the cavity resonance problem, the interesting values of y are comparable with n and the full expression (44) must be used. However, even in these cases, it is important to know that Iv’(y)/v(y)l is of the order of one. 7. DISCUSSION OF THE CAVITY RESONANCES In order to solve (43) for cases of interest, it is written in the following valent form :

equi(47)

Since 6 < 1, a reasonable the denominator. Thus,

approximation

to (47) is to neglect the second term in

a,=--_,

iA

(43)

s

where A=

From (37) and (48), it is seen that x = x, = W,a/c = [;F;A;;J1’2,

(49)

which yields the complex resonant frequencies We in the absence of coupling. It is not quite an explicit equation for o, since A/s itself is a function of o. However, in the case of small damping, IMI> 1 and cc), is almost real. Thus, inside the radical of (49), the following approximation is valid: A --N 6 -

C Mw,'(b

-

a)'

where M=

qpo2(~ + iu,‘)d (Y + iwnry

+ co,2

and 02,’ = Rew,zjconI.

(59)

where

Equation However,

(Sl), used in conjunction with (LX), may he solved as a cubic in (o,,:‘). in most cases of practical interest. 1~ ,’ CO,>. and, t,hus.

This interesting result indicates, if the electSron densit,y of the shell is suficientl>~ e-i 1, and the resonant frequencies are given by high, that Re SI,, (‘J,i’

; In(n

l)]“’

for >/

I . 2. 3.

or. if n == 6370 km (the radius of the earth). f,L’ ~= (0

,‘k?ls

i.50

I?%(?7

I)]“’

c;s.

Equation (53) above shows that the finite elect#ron density of the shell either lolvrrs or raises the resonant frequencies. depending on whet’hcr Y is greater or smaller than OjC,respectively. The damping of the oscillations is dependent on the imaginary part of ~1))~.Thus. W,,”

In7 (I),,

:, In(n, 7 I)j’:2 lm 1I Lm

x,,] ‘I2 . Y.,,:

(54)

where (55)

Since shock excited oscillations will vary with time approximately to iwn’t), the damping will also be positive, corresponding Again, some simplification is obtained when 1’ .J CO,,.Then

as exp ( -mill decaying WAWS.

This result clearly shows that the damping is increased by the presence of the 11.c. magnetic field. However, if the d.c. magnetic field is sufficiently small that CO,. v. the damping is proportional to Im cl,{ :_ _&___.~,

. 02doI,

which illustrates a linear dependence

on the collision frequency.

(!ii)

Cavity

resonances

for a spherical

8. FINAL

earth with a concentric

anisotropic

shell

89

REMARKS AND CONCLUSIONS

The foregoing discussion is based on a solution of (47) with the term proportional In many cases this is justified provided the product d2N2 is to N2 neglected. reasonably small compared with unity. This condition is most readily violated when the d.c. magnetic field is very strong. In this case, (47) should be used without approximation. A more serious limitation is the assumption of the constancy of M and N. For example, the effective verical component w, of the gyrofrequency will vary approximately as the cos 6 when the coordinates are chosen coaxially with the geomagnetic axis. Thus, M will be maximum at the equator and be reduced at the poles with a smooth variation at intermediate points. On the other hand, the factor N will vary from zero at the equator to a maximum at the poles. To understand the significance of these variations of M and N requires solving a rather involved boundary value problem by the method suggested earlier. It appears that some simplification is achieved if it is recognized that, for the cavity resonance problem, the cross-coupling it is really only factor N plays a minor role. Thus, for a good approximation, necessary to solve the decoupled TM mode equation with impedance factor A regarded as a function of 8 and 4. In future papers, it is hoped to report progress on this aspect of the problem. The principal conclusions from the present study are that (a) the vertical component of the magnetic field appreciably increases the damping of the cavity resonances, (b) the coupling between TM and TE modes plays a small role, (c) the resonant frequencies are lowered by the presence of appreciable loss in the concentric shell and modified to some extent by the earth’s magnetic field. Such conclusions are not inconsistent with experimental data such as described in the comprehensive review paper by GALEJS (1963). 9. REFERENCES BUDDEN K. G. CHAPMAN S.

1961 1956

GALEJS J.

1963

HOFFMAN W. C.

1964

THOMPSON W. B.

1963

WAIT WAIT WAIT

1961a 1961b 1962

J. R. J. R. J. R.

WAIT J. R. WAIT J. R.

1963 1964

Radio wazte~ in the Ionosphere. Cambridge Univ. Press, Cambridge. The electric conductivity of the ionosphere: a review, Nuovo Cimento 4, (Supplement), p. 1385. Terrestrial extremely low frequency propagation. To be published in Proceedings of NATO Advanced Study Institute on Low FrequencyElectromagneticRadiation,BadHomburg,Germany. Propagation of Radio Waves at Frequencies below 300 kc/s. (Edited by W. T. BLACKBAND), Pergamon Press, Oxford. A layered model approach to the earth-ionosphere cavity resonance problem, Ph.D. Thesis, Dept. of Geology and Geophysics, Mass. Institute of Tech. Appl. Sci. Res. Sec. B. 8, 397. J. Res. Nat. Bur. Stand 6!jB, 137. Electromagnetic waves in stratiled media. Pergamon Press, Oxford and Macmillan Co., New York. J. Res. Nat. Bur. Stand 67D, 297. Canad. J. Phys. 42, 575.