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RADIO SCIENCE Journal of Research NBSjUSNC- URSI Vol. 69D, No.8, August 1965

Schumann Resonances 1

Janis Galejs

Applied Research Laboratory, Sylvania Electronic Systems, a Division of Sylvania Electric Products Inc., 40 Sylvan Road, Waltham, Mass. 02154

(Received February 3, 1965)

This paper revi ews the curre nt work in the fi eld of Schumann resonances and di scusses the topi cs of waveforms and frequency estimates, resonance frequencies and Q fa ctors, source dI strIbutIOns and noise spectra. diurnal and seasonal c hanges of power spectra, and .variations of resonance freque ncIes. Observations of th e above ph e nomena a re corre lat ed WIth th eoretI cal cons Id eratIons.

1. Introduction This model permits a close reproduc tion of the noise spectra measure ments in the resonance region, and The resonances in the spheri cal shell bet ween the it also provides an agreement with measured ELF earth and ionos phere that are due to azimuthal waves attenuation rates [Chapman and Macario, 1956; have been considered in two recent s urvey papers J ean et a1., 1961). [Galejs, 1964a; Wait, 1964a]. This work contains Models of an a ni sotropi c ionosphere are difficult to experimental data as well as the required theory. apply to the problem of earth-to-ionosphere cavity The basic theory of the resonances has been re vi e wed resonances because of the variations of the magneti c by Wait [1964b). In the present summary the fi e ld vector along the s urface of th e earth. The prop­ emphasis will be on correlation between measurements agation parameters can be estimated in the presence and available theory. There will be no extensiv e of a horizontal magneti c fi eld component from the analytical developments, but frequent references will work of Galejs and Row [1964] and of Galejs [1964b). be made to literature. The earth-to-ionos phere cavity resonances have been Based on theoretical considerations, Schumann analyzed for a nonhomogeneous ionosphere in th e [1952a, b, 1957] postulated resonances of the earth­ presence of a radial (vertical) magneti c fi eld by Thomp­ to-ionosphere cavity. Koenig [1958, 1959, 1961] son [1963] and Galejs [1965). A thin-s hell approxi· obtained the first experimental indication of Schumann mation of the ionosphere with a s uperimposed radial resonances by observing noise waveforms in the out­ magne ti c fi eld has been conside red by Wait [1964c). put of a narrow band amplifier. Detailed freque ncy Some other work of Wait [1963a, b], although intended spectra of this noise were first obtained by Balser and for a different frequency range, can also be applied Wagner [1960]. Other measurements have been to the earth-to-ionosphere resonance problem in the reported by Fournier [1960]; Benoit and Houri [1961, presence of a radial magnetic field. 1962]; Lokken et a1. [1961, 1962]; Polk and Fitchen Additional references are available in the compre­ [1962]; Gendrin and Stefant [1962a]; Balser and Wagner hensive bibliography by Brock-Nannestad [1962]. [1962a, b, 1963]; Rycroft [1963]; Chapman and Jones (1964). The fundamental theory of Schumann resonances 2. Waveforms and Frequency Estimates [Schumann 1952a, b, 1957] is discussed in a book by In wide band recordings of atmos pheric signals an Wait [1962]. Raemer [1961a, b] considers the observ­ ELF component or slow tail follow s the initial VLF able noise spectra as the response of the earth-to­ component of the transient [He pburn and Pierce, ionosphere cavity due to li ghtning flashes all over the 1953; Liebermann, 1956a, b; Tepl ey, 1959; Pierce, world, but the homogeneous sharply bounded iono­ 1960]. An oscillatory structure of the signal becomes sphere model of Rae mer introduces high losses, and he noticeable by passin g it through a band-pass filter. does not succeed in reproducing the spectral measure­ An example of a noi se recording in the output of a 2 to ments of Balser and Wagner [1960]. Galejs [1961a, b] 30 cis band-pass filt er [Lokken, 1964, private communi­ uses an isotropic ionosphere model of exponentially cation] is show n in fi gure 1. The two horizontal mag­ increasing conductivity which is based on measured netic fi eld components have been observed simul­ or calculated characteristics of the lower ionosphere. taneously in Canada and in Antarctica. There is a I Paper present ed at th e UL F Symposiu m, Buu ld er. Co lo. , Aug. 17- 20, 1964. correlation between the hi gh intensity ELF noise bursts 1043

------observed at the various sites. A recording of one of d ' N-S the waveforms is shown in figure 2. The darkened .. " ...... areas indicate resonance frequencies around 8, 14, 8/ 22 / 61 ~ and 20 cis. Noise waveforms in the 5 to 20 cis band E-W ~E-W have been reported by Polk and Fitchen [1962] and Polk [1962] . Figure 3 shows sample recordings of the outputs of two coils the axes of which are oriented 'h ' , ':wi" in the north to south (N-S) and east to west (E-W) N-S 'WCIf/fIININIW• "'M; ... directions. Frequency of these waveforms has been estimated by counting the cycles between the second 9/ 20 / 61 10/9/ 61 markers shown on the records. The average fre· E-W E-W quency computed in this procedure should be higher FIGURE 3. Noise recording in the 5 to 20 cis band.

x than the resonance frequencies of a spectral analysis GREAT WHALE RIVER, P.Q. [Galejs, 1962a, 1964al Noise recordings made simul· y taneously in Germany and in Rhode Island have been reported by Keefe, Polk, and Koenig r1964].

x WESTHAM ISLAND, S. C. y 3. Resonance Frequencies and Q Factors ~ ____...... ".. ___~""w,~. , ~" ..... , •• ,." 3.1. Definitions A resonance frequency and Q factor may be deter· x mined from spectral measurements. Analytically it is also possible to compute the power spectrum of the noise waveform. However, it is more convenient to relate the resonance frequency and the Q factor to 36 37 41 42 the analytically determined propagation parameters. " 1639Z 10 FEBRUARY 1963 The radial electrical field due to the vertical dipole FIGURE 1. Noise recording in the 2 to 30 cis band. excitation can be represented by

(1)

where

v(v + 1) = (koaS)2. (2) _ : . For a wave that propagates in the spherical shell be· 1635Z tween the earth and the ionosphere S is defined as the ratio of the complex wave number k to free space wave number ko(S = klko). The real part of S can be seen to be inversely proportional to the phase velocity, and the imaginary part of S is proportional to the at· tenuation constant. This parameter S is determined from solving the appropriate boundary value problem of the earth, the air space and the chosen ionosphere 1640Z model [Wait, 1962; Galejs, 1964aJ. The resonance frequencies of the nth mode are determined from the minima of sin V1T in (1) [Galejs, 1962a, 1964a] which gives

f, = 7.5 v'n(n+ 1). 10 15 20 25 (3) n ReS FREQUENCY - cis

FIGURE 2. Recording of the noise waveform X of West ham Island, Typical values of ReS are 1.4 to 1.2 for frequencies in B.C. the resonance region. 1044 The cavity Q may be determined as a ratio between tensor is of the form the stored energy and energy loss per cycle or simply by the width of the resonance curves. As s hown in -0'2 the appendix, in the present problem the electri cally stored energy We differs from the magnetically stored [0'] = : 2 0'1 (7) energy Wh, and the Q definiti on should consid er the 0 :J0'0 sum of the average stored energies We and Wit. This r' expression for Q is Equations for computing the conductivity components (ReS)2 + 1 are available in literature [Galejs, 1964a and b] and Q (4) typical conductivity profiles are shown in figure 4 but 4ReS ImS some uncertainties still apply to such models [Van­ zandt' 1964]. The above computations consider the ion effects, but assume the operating frequency to be When considering only the magnetically stored energy zero. These profiles can be further approximated by Q is given by straight lines through the lower ionosphere in the semilogarithmic plots of figure 4, which corresponds to Q= 1 2 ReSlmS (5) an expoential height variation of the conductivity com­ ponents. In a further simplification the anisotropy may be ignored (

Q= ReS. (6) O'(z) = 0' (zo) exp [/3 (z - zo)] (8) 2ImS where zo = 60 km and the day and ni ght models are The last expression can also be derived from the con­ characterized by O'(zo) = 4.63 X 10- 8 mho/m and cept of complex resonance frequencies [Wait, 1964a] ,8 = 0.308 km - I and by O'(zo) = 6.5 X 10- 10 mho/m or by considering the half-power band width of the and ,8 = 0.44 km- I , respectively . resonance curve. All three Q definitions yield the same results as ReS approaches unity, but they will differ near the lower resonance frequency of the earth ionos phere cavity. Equation (4) will be used in the prese nt calculation. Experimentally determined Q values that are esti­ mated from the half-power bandwidths of the reso­ 10 nance curves neglect the effects of adjacent resonances '0 - "i FOR flECTRON S ONLY

10 and of near-field noise that will tend to add to the -- "i FOR ftfCUONS '0 background level. Hence, the half-power level is AND IONS DAY TI ME estimated too low, the apparent half-power bandwidth 10- 1 10.1 is larger and the estimated Q factor may be too low. N IGHTTI ME 10.7 10-2

3.2. Characteristics of the Boundary ~ ~ 10.3 ~ 10-3 \1 ~ > 0 In the theoretical models that are advanced for ex­ 10.4 ~ 10-4 plaining the resonance frequencies and the Q-factor > .- ;; > measure ments [Wait, 1962; Galejs, 1964a; Wait, 1964a] ~ 10-5 ;; 10-5 it is necessary to consider the boundary properties ~ " z~ ~ of the spherical shell. 8 10"6 , Z 10.6 / \'" ,, 8 In the frequency range of the resonances the dis­ , / \ \ " , " , "- placement currents of ground are negligible and the 10.7 , 10.7 \ "- 11'1" \ "-' ,,- ground conductivity O'g= 10- 3 to 1 mho/m is much 1 \ 10. 8 -

Compressibility effects of the ionosphere have been f IN CIS 10 30 100 considered by Seshadri [1965b] using the model of a perfectly conducting flat earth and a sharply bounded DAY WITH h = 4 5 KM, NIG HT WITH h = 60 KM 8 .5 6 . 7 homogeneous isotropic and lossless ionosphere. The compressibility of the ionosphere does not affect the DAYTIME ONLY WITH h = 45 KM 6. 5 5 .4. 5 .2 DA Y WITH h '" 45 KM, NI GH T WITH h = 60 KM , wave propagation below it for frequencies in the range COSMI C RAY IONIZATION 5.5 6.4 of Schumann resonances and in the ELF band. How· ever, the compressibility should be considered for MEAS UR EMENTS waves that propagate inside the ionosphere [Seshadri, 1965a].

1046 3.6. Magnetic Field Effects consider electrons and ions at an operating fre­ quency of 20 cis, as indicated in figures 3 and 4 The magnetic field effects for a homogeneous sharply of Galejs [1964b]. In the isotropic ionosphere bounded ionosphere model have been discussed by models (Fo remains unaltered, (FI = (Fo, and (F2 ~ O. Wait [1962] in a quasi-longitudinal approximation for The calculated resonance frequencies f" are li sted in a superimposed radial magnetic field and also for a table III. The isotropic day or night models exhibit transverse magne tic field (east-west or west-east propa­ too high resonance frequencies, but the average for gation along the magnetic equator). the anisotropic day or night models gives nearly The propagation parameters have been determined correct results, although the first resonance occurs by Galejs and Row [1964] and Galejs [1964a, 1964b] near 7.7 cis. At night the various anisotropic models in the presence of a horizontal magnetic field. These give different results, while their results are nearly the data provide an indication of propagation character­ same for daytime or for the isotropic models. The istics of the waves near the magnetic equator, but calculated Q factors are listed in table IV. At daytime they do not apply directly to the earth-ionosphere the Q factors are lowered slightly due to anisotropy. resonance problem, where the H vector is nearly hori­ The losses are principally due to absorption in the zontal only over a small surface portion of the earth. lower D region where the collision frequencies are The earth-ionosphere cavity resonance frequencies high and the magnetic field has only small effects. At have been considered in the thesis of Thompson nighttime the anisotropy reduces the Q values dras­ [1963] for a radial magneti c field. He uses a multi­ tically, and the ions have a considerable effect. For layer ionosphere model that approximates the con­ the isotropic models the energy remains below the tinuous altitude vari ation of the conductivity tensor. ionosphere, the losses in the lower ionosphere are small The azimuthal fi eld vari ation is expressed in terms of and the Q values are high. For the anisotropi c models the Legendre polynomials Pn (cos 8) for the nth mode. some energy penetrates the ionosphere and escapes The wave equa tion has four different solutions for from the earth to ionosphere cavity. Hence the Q each of the ionosphere layers, but it has onl y two fi gure is low. solutions for the space between the ionos phere and the perfectly conducting ground surface. The solu­ TABLE III tion of the proble m and the resonance condition for RES ONANCE FR EQ UENC I ES OF THE SPHERI CAL SH ELL BETWE EN TH E EARTH AND IONOSPHERE I N THE PR ES EN CE OF A RAD IAL MAGNETIC FIELD the mode n are obtained after multiplying a seque nce of 4 X 4 matrices. In these computations the di splace­ I ONOSPHERE MODEL ~ 1 2 , 4

ment currents of the ionosphere are neglected. The ~ DAY, AN I SOTROPI C <.> 7 . 8 13 .8 19 . 8 26 computed resonance frequencies are in agreement ~ DAY, I SOTROPI C 8.1 14 .2 20. 4 26.6 with measureme nts, but the Q values are in the range ~ " of 15 to 20, with the excepti on of ni ghttime ani sotropi c g NI GHT, ANISOTROPl C £ L.+ IONS F "" 2 0 7. 6 15 .2 23.2 29. 5 solutions, where Q = 7 to 10. The lowering of the ~ nighttime figures by the magneti c fi eld has been ::: EL.+ IONS F r O 7 . 7 13 . 8 20. 1 26. 9 Q w attributed to wave penetration through the ionosphere ~ EL. ONLY F • 0 7.8 14 . 1 20.7 27. 8 and to energy thus escaping from the earth-ionos phere cavity. ~ NIGHT, I SOTROPIC 8.6 15.3 21. 3 21 .4 Another solution to the proble m of earth's MEASUR EMENTS 8 14 . 1 20. 3 26. 4 ionosphere cav ity resonance in th e presence of a radial magne ti c field is given by Galejs [1965]. The azimuthal fi eld variati on is expressed in terms of TAB LE IV the Legendre functi ons P v ( - cos 8) , and the solution Q- FACTORS OF fARnt-IONOS PHERE CAV ITY RESONAN CES I N THE PRESENCE OF A RADIAL MAGNETIC of the wave equation is set up so that the four solutions FIELD of the ionos phe ric layers where (F > WEo degenerate IONOSPHER E MODEL ~ 10 30 into two soluti ons for the lower . ionospheri c layers where (F < < W Eo and for free space. After comput­ DAY , AN I SOTROPI C 6 . 8 5.6 in g the surface impedance at the lower boundary of the ionosphere, the propagation parameters are deter­ DAY, I SOTROPI C 8 . , 7 mined from the solutions of the modal equation for TM NI GHT, ANISOTROPIC EL. + IONS F "" 20 1. 7 6.6 modes, althought the coupling between TE and TM modes is considered in the processes of computing the EL. + IONS F "" 0 2.0 2 .7 surface impedance. Only the vertically polarized TM EL. ONLY 2. 8 4. 5 modes are assumed to propagate below the ionosphere. F "" 0 Such assumptions have been justifi ed for a single layer NIGHT , I SOTROPIC 8 . 8 9.' ionosphere model at hi gher frequencies [Wait 1962, p. 269] and for a thin ionospheric s hell al so for fre­ MEA SUREMENTS 4 6 quencies in the resonance range [Wait, 1964c]. In the numerical calculations the anisotropic The previous calculati ons have assumed a constant ionosphere models are those of figure 4 and further radial magneti c field (S pole) all over the globe. A calculations are made for ionosphere models which reversal of the static magneti c field (N pole) causes

1047 77 1-0250-65-2 I

------J negligible differences in the propagation parameters or 1.4r-.------, in the values of In and Q. Hence a model using the ----RAEMER (1961) average of the radial magnetic field may provide a 1.2 ----- RA EME R (1963) first order estimate of the anisotropy effects. The • -- • -RYCROFT (1 963) present calculations assume the radial magnetic of the ~ 1.0 Z polar regions to be extended over the whole surface of :J the globe and the anisotropy effects should be too 2! 0.8 pronounced. The analytical results are highly 3 dependent on the detailed structure of the nighttime w '" anisotropic ionosphere, and further studies of it may , 0.6 be in order. '"

~0.. 0.4 4. Source Distributions and Noise Spectra 0.2 Most of the ELF energy is of terrestrial origin and is caused by thunderstorm activity, although there are o 2~~~-.1~0--'174-~18~~2~2-~26~~3~0-~374-~3~8--d42 some high latitude events of an apparent extra­ FREQUENCY -- cis terrestrial origin. Such emissions of approximately 800 cis ha've been detected near Kiruna, Sweden [Egeland, 1964] and of 14 to 17 cis on Kerguelen Island FIGURE 7. Estimated Lighting flash spectrum. in the Southern part of the Indian Ocean [Gendrin and Stephant, 1964, private communication]. The Spectran recording of the signals received at Kerguelen Raemer [1961a, b] has estimated the power spectrum is shown in figure 6. The upper trace shows a normal of the dipole moment in a median lightning flash Ids (or signal with resonances near 8, 14, 20, and 25 cis. the first derivative of the electrostatic moment dMldt) There is an additional signal of 14 to 17 cis on the two based on statistics of lightning strokes compiled by lower traces. When neglecting the presence of such Williams [1959]. Later work of Raemer [1963 private extraneous signals the terrestrial noise spectra can communication] has lead to a revised estimate. These be calculated after estimating the dipole moments of spectra are shown in figure 7. Pierce [1963] has com­ the sources and using propagation parameters that are puted the magnitude of the frequency spectra of the appropriate to the selected ionosphere model. second derivative d 2 Mldt 2 , and Rycroft [1963] has calculated the frequency spectra of dM/dt using a similar representation of the lightning moments. The square of the frequency spectrum computed by Rycroft [1963] is also indicated in figure 7. An equivalent power spectrum for the dipole moment 30 ~ 20 t of terrestrial noise sources has been deduced by Harris and Tanner [1962] from the measurements of ~ 10 Balser and Wagner [1960] by estimating the complex o propagation constant within this frequency range in a "trial and error" approach. Harris and Tanner [1962] do not require an ionosphere model or any knowledge of lightning waveforms and their statistics for obtaining this equivalent power spectrum. After determining the source spectra and the propa­ gation parameters the power spectrum of the received noise may be computed from

G{iw) = g(iw) 1v (v + 1) 12 8(akoh)2 1 v + t 1 [cosh (217 1m v) - cos (217 Re v)] . { sinh [21m v (17 - 8) ] (- 1m v)

_ sin [(2 Re v+ 1) (17- 8) -(17/2)]}82 Re v+ ~ 8 1' (9)

where the v is given by (2) and where the sources are assumed to be uniformly distributed in the 8 interval of 8, < 8 < 82. The derivation of (9) starts from (1) and Legendre functions Pv(- cos 8) are replaced by an asymptotic expansion. There are also equivalent spectral representations in terms of Legendre poly­ FIGURE 6. Spectran recordings of noise waveforms at KergueLen IsLand. nomials [Raemer, 1961a, b]. 1048 25r----,-----,----,----~--__,.--~ --RYCROFT 4.0 _ __ BALSER AND WAGNER -- THEORETICAL --CALCULATIONS ...~ ---- EXPERIMENTAL 10 ~ 3.2 . I w > 1 ;1'\, ;:: I.1i i 1 2.4 15 i fl · / /1 I ! / 1.6 // \ I 1 i'· ..· .. ~ 10 1 \ I \ , I I l.s I, \ '7 f\ \ . I I \ ,\ 0 \'::;11 32 36 I \ / 0 4 12 16 20 24 28 \ I \ / FREQUENCY- cis '- ..... /

FIG URE 8. ELF Noise spectrum-effective ionospheric height 10 15 10 25 30 assumed constant. FREQUENCY - cis FIGURE 10. Noise spectrum/or 47".;; e.;; 128°.

25 -- • -- • BALSER AND WAGNER ---IPSWICH __ -- -BENO IT AND HOURI - - -- - PARIS 20 -- - - TROMSO /'\ . '" I \ 15 ~ \ \ ~... \ :5 10 w '" \ \. \

10 15 20 25 30 FREQUENC Y - cis

F IGURE 9. Noise spectrum/or 38°.;; e.;; 72° .

'" , The noise spectrum of Raemer [1961b] has been \ ,\.,....-../. /' / ...... used for computing the observed noise spectra in ,_ ...... \ , conjunction with a sharply bounded ionosphere model. ~'r--T- __ ...... ·/1...... This spectrum is shown in figure 8 and is seen to exhibit 1":.::...... -,.;..-_--- ""20"'C",/2:, :::::.- ...... _ ...... --<." __ ' ...... excessively damped higher resonance peaks. The exponential ionosphere model [Galejs, 1961b,

1962a, 1964a] produces a better agreement with meas­ 12 16 20 urements as seen in figures 9 and 10. In this work UNIVERSAL TIME propagation is assumed to be uniform around the sur­ face of the earth and the propagation parameters FIG URE 11. Diurnal variation 0/ the resonance modes at Ipswich, Mass., February 1961 (ve rtical electric field), Chambon-la·Foret represent the average of day and night conditions. (near Paris) July 1962 and at Tromso (Norway) April 1962 (hori· The sources are assumed to be uniformly distributed zontal magnetic field). over the () interval indicated in the figures (() is defined with respect to the observer).

5. Diurnal and Seasonal Changes of at the first three resonance frequencies are shown in Power Spectra figure 11 together with similar measurements by Gendrin and Stefant [1962a]' Diurnal variations of Balser and Wagner [1962a] have recorded diurnal the fields near the two lower resonance frequencies variations of measured noise spectra. Their meas­ have been measured at Kingston, R.I., by Polk, Huck, urements of the diurnal variations of the power level and Yu [1963, private communication]. The maxi- 1049

------mum signal intensity occurs between 1500 and 1800 sphere (2n1r) and has also experienced a phase shift UT. In an amplitude versus time plot the signal peak of (-1r /2) radians at each traversal of the source point is broader during summer than winter. Seasonal and or its antipode_ This physical interpretation of (13), diurnal variations of the noise power in the ELF fre· which is equivalent to (1) for v= const., per~mits a quency band below 35 cis have been described by heuristic consideration of the v differences between the Wright [1963] without singling out the individual day and night hemispheres. The multiplicative v fac­ resonance modes. The measurements are made at tors in front of the n summation are assumed to be con­ Byrd Station, Antarctica (800S, 1200W) and show peak stant and equal to 'j) which is intermediate between intensities near 1000 UT during spring and summer the day and night v values. The () factors of the ex­ and near 2300 UT during most of the year. ponentials are replaced by integrals in order to The gross features of the spectrum, such as the account for the v variation seen by the circular wave number of peaks and their approximate time of occur­ front as it propagates in the (}-direction with respect ance, may be explained qualitatively by noting areas to the source. The amplitude changes at the day of globe with particularly high thunderstorm activity and night boundary can be also considered for each and by measuring the polar angle () from these source traveling wave by assuming that the energy of the regions to the observation point. The vertical elec­ wave remains constant after passing the day and night tric field E I' and the horizontal magnetic field H 1> of boundary. This results in decreased amplitudes for the mode n exhibit the () dependence of the larger ionospheric height of the night hemisphere. In the numerical work the surface of the earth was E,. - Pn (cos(}) (10) subdivided into 10° cells. Each cell had a constant d surface density of noise sources but a variation was Hq, - d(} Pn (cos(}), allowed from cell to cell. The plots of thunderstorm (11) days over the world were used to set relative values where Pn(cos (}) is a Legendre polynomial of the order of lightning activity for each of the cells, by assuming n. A particular geographical region will contribute that the distribution of squared dipole moment per unit area is directly proportional to the thunderstorm to the measured values of E,. or Hcb in the mode n if the right hand sides of (10) or (1) are sizeable for those day plots according to the Handbook of Geophysics [1960). The diurnal variations were accounted for values of () _ Such considerations have been reported by a multiplicative factor in terms of local time at by Balser and Wagner [1962a] and Gendrin and the source [Williams, 1959]. The final equation Stefant [1962a]. used in the spectrum calculations is similar to (9), More detailed calculations have been carried out by although it is algebraically more involved [Abraham, Abraham [1964, private communication; Galejs 1964a] 1964, private communication). who used available estimates of geographical distribu­ The diurnal variations of the noise power at the tions of the noise sources and who calculated the day first three Schumann resonances have been computed and night propagation effects after expressing the fields using the exponential ionosphere model of figure 5 as a sum of recirculating azimuthal waves. and are shown in figure 12. The calculations have dis­ In the expression for the electric field component regarded the nOIse sources near the antipode where E,. of (1), the Legendre function is replaced by the the usual asymptotic approximation of the Legendre leading term of its asymptotic expansion. The func­ functions is inaccurate. Figure 12 shows a qualitative tion sin V1r is expanded into agreement with the measurements of Balser and

00 Wagner [1962a], that are shown in figure II. (sin V1r)-1 =-2i ~ e i(2n+ 1)"7T (12) An example of seasonal power variations is shown n=O in figure 13. It indicates a decreased noise power which is valid for 1m v > O. This results III [Abra- near the higher frequencies and/or during winter ham, 1964, private communication] months. iJ,jr Based on measurements at one location of the earth E1'=~ v(v+l) e it is also possible to predict the noise levels at other 2wEha2 V21r(v+ J)sin () locations using this procedure of noise spectrum calculations. The noise that is due to a worldwide ~ {e i(v + ;)(O+ 2117T )-i2nT+e i(v+ })[27T(n+t)-ol-i(2n+t)~ }. distribution of thunderstorms depends on the fre­ n=O quency variation and also on the absolute value of the power spectrum of the mean dipole moment of (13) the source g(iw, (), depende nce may The first term of the summation represents a wave be estimated from the thunderstorm frequency maps which has traveled the direct distance from the source as was indicated before, which leaves the absolute to the observation point ((}) plus n full circles around value and the frequency dependence of g(iw) as an the sphere (2n1r), while experiencing a phase shift of unknown. Assuming that g(iw) is the same for all (-1r / 2) radians at each traversal of the source point noise sources, it may be determined by making the or its antipode_ The other term of the summation calculations to agree with measurements at one par­ represents a wave which has traveled toward the ob­ ticular location. This gives a uniquely determined servation point across the antipode of the source noise distribution over the surface of the earth and (21r - ()) and has made furthermore n circles around the the accuracy of this model can be checked by com- 1050 ------

8 cis -- 8 -70

~ - 80

-90 --- } MA LT A o 2100 LOCAL TIME

I.J..JN - 100 9 ;;: ---} A LASKA o 1000 LOCAL TI ME u B - NORMALI ZED WIT H RESPECT TO 80ULDER COLORADO - 110 '"~ (D ECO MEASUREMENTS) u- NORMALI ZED WIT H RESP ECT TO LONG ISLAND N . Y. (R CA ME ASUREME NTS )

FREQUE NCY - CI S

FI(;UHE t 4 . COIII /)(I r i son of cll lc"llIled fiel d sl rell;:lh 's dlllll lV ilh .Julv 196.3 III PII .\IIrelllellls br DECO lind RCA.

UNIVERSAL TIME paring the calculations with m e a ~ ure m e nt s at other locations. In an example, measure me nts by DECO F IG URE 12. Calculated diurnal variation of noise power at th e peak of the reson ance morles - winlertime - 80S tOil , Mass. at Boulder, Colo., [Maxwell, ]963, private communi­ cation] and by RCA at Long Island, N.Y. [Powers, 1964, private communication] , we re used to determin e g(iw). The cal culations by Abraha m [1964, private communication] based on th ese values of g(iw) have been compared with simultaneou s meas ureme nts made in Mali a and Alaska [Maxwell , 1963, private communication]. As seen from fi gure 14 the analytic curves agree in s hape with the meas ure ments, but the normalization with respect to Boulde r giv es somewhat highe r predi ctions of the noi se level. This may pos­ 50 sibly be caused by summertime thunderstorm activity in the mountains nea r Boulder, Colo _. that would giv e significant near-fields and a hi gh estimate for the func­ tion g( iw). The calculations are extended above the

(Il 40 region of the earth-ionosphere cavity resonances to "0 show that more significant noise level changes occur at higher frequencies. In the calculations leading to figures 12 through 14 w the day and night effe cts are considered most ac­ ~ 30 curately if the day and night boundary is located sym­ g metrically with respect to source (i_e ., source at local noon or midnight). For nonsymmetrical day and night boundaries it is difficult to estimate the effects 20 of wave front distortion_ When neglecting such distor­ tions (or assuming propagation strictly in the 0- direction), this method can account for v(O) variations, which may be caused by ionosphere parameter changes or by changes of the static magnetic field along the 10 surface of the earth. The assymptotic expansions of P v< - cosO) used in (13) or (9) can be modified to increase their accuracy for observations near the source or its antipode [Abraham, 1965]. o LLWU~ __-L~~-L~~L---~-L-L~~~ 5 \0 20 50 100 200 500 I K FREQUENCY - cis 6_ Variations of Resonance Frequencies

FIGUHE 13 . Calculaled noise spectrum for local noon in northern Diurnal frequenc y variations of the earth-ionosphere hemisphere. cavity modes have been reported by Balser and Wagner 1051

------[1962b], Gendrin and Stefant [1963], Rycroft [1963] and by Chapman and Jones [1964b]. The resonance frequencies vary by about ± 0.2 to 0.3 cis , but the changes are not simultaneous for the different modes; 26"t26 ' 0

L~080 ~ j a low resonance frequency of one mode occurs simul­ 0 j 25,5 taneously with a high frequency of a different mode. , ~~'f 1 I i \ " 19.5, I I l::oo 1 , 26,51 0 0 " "0 0 0 0·... • •• • I 26 r .'O.'--''----...JL----''----''• ..:·--''-.-:.· .... ~---tI 25,5 : '~:l~----'l~eo6 --;-0 ~- 1 I i1 13,5 I I , -s 20,5 ) . 00 0 o 0 >-I 20"" 0 0 a 0 a ••••• 0 a O. 0 0 ~ 19,5 00 0 0 ,A tI " I ~ j I J-----,. ~ : I , 12 16 UNIVERSAL TIME

FIGURE 17. Nuclear effects on earth· ionosphere cavity resonances. tI I Measurements by Gendrin and Stefant. I

A high altitude nuclear explosion has been shown to tI I I affect the earth-to.ionosphere cavity resonances by I 16 24 simultaneously lowering all the resonance frequencies UNIVERSAL TI ME [Gendrin and Stefant, 1962a and b; Balser and Wagner, FIGURE 15. Diurnal frequency variations measured by Gendrin 1963]. Tepley et al., [1963] noted also an increase of and Stefant. , signal strength for observations near the conjugate point of the explosion. The lowering of resonance frequency shown in figure 17 has been attributed to worldwide lowering of the effective ionosphere in a model of a single layer isotropic ionosphere [Gendrin 15 .0 and Stefant, 1962b]. The depression of the resonance 0 0 frequency can be also explained by assuming an ex­ ,0 0 0 0 0 ponential daytime ionosphere to be effective around 0 0 0 0 0 0 , 0 the globe [Row, 1964, private communication]. As an 14 . 0 -1------, ------0 , illustration the daytime resonance frequencies of ,0 0 table 1 compare with the measured resonance fre· , 0 0 0 quencies of figure 17. There are VLF observations 0 13 .0 [Zmuda et aI, 1963] that can be explained by a lower· V> "- U ing (or increasing electron density) of the D-region I 0 boundary. Brady et al., [1964] interpreted VLF di· u>- ~ urnal phase shifts to show that some enhanced D· :::> 12.0 region ionization persists for several days following g 8. 6 ~ the event.

00 The lowering of the resonance frequency following 0 • a nuclear explosion has been attributed by Chapman 8.2 . • 0 0 • • and Jones [1964a] in their two-layer ionosphere model "" 0 0, solely to a decreasing conductivity of the E-region. ------.~.---- Mter measuring a change of the resonance frequen. o o .. o cies following a nuclear explosion, it is possible to 7.8 , 0 advance simple ionosphere models for explaining a o particular change. However, no attempt has been reported to predict resonance shifts for different 7.4L-~-~-~-~-~-~-~~~~-~ nuclear yields and detonation altitudes. This rather 0600 1000 1400 1800 2200 0200 complex problem requires a quantitative evaluation UNIVERSAL TIME of the various nuclear mechanisms that may lead to FIGURE 16. Diurnal frequency variations measured by Rycroft. ionosphere perturbations. If such information is 1052 available, the computation of the resonance shifts lossy walls is is relatively simple. It would be also of interest to investigate the effects of sudden ionosphere di s· 1 I I E x H * . ds Qi (14) turbances (SID) and to di scriminate between SID's WEo I I IE· E *dv +w/-to I I I H· H*dv and nuclear effects. 2 2 The resonant frequency changes can be considered more accurately with nonuniform models of the propa· gation geometry. A perturbation method [Wait, where the surface integral is extended over the cavity 1964b] or numerical techniques [Madden and Thomp· walls and the volume integral covers the inside of the son, 1964] have been used for solving such problems. cavity. The ground surface is assumed to be a perfect Madden and Thompson [1964] compute the local conductor and E8 is related to Reb at the ionospheric surface impedance and propagation constant for a boundary by the surface impedance Zs as Ee=ZsHeb ' uniform cylindrical shell between the anisotropic However Ee is assumed to be small relative to Er for ionosphere and the earth using matrix multiplication vertically polarized waves. Equation (14) is rewritten techniques. There is a smooth variation of the as parame ters between polar and equatorial regions and also between day and night hemispheres. Tnese 1 local surface impedances and propagation constants (15) are assumed to characterize a spheri cal transmission Qi line of nonuniform parameters. The resonance fre· quencies and Q factors are computed numeri cally using a lumped parameter approximation of the two· 2ReZs I IHeb l2 sin ede (16) dimensional transmission line. This model has been = -;;;h J[Eo1Er1 2 + /-toIR eb I2] sin ede applied to explain the diurnal frequency variations and also the frequency changes due to a nuclear where ais the radius of earth, h is the ionospheric explosion. height and where the cavity is assumed to be uniform. For excitation by a radial electric dipole of moment Idr the fi eld component E,. and Heb follow from (21), (61), and (62) of Galejs [1964a] as 1) E.,.= ---,-----,--ildr n(n+--'-- 1-:':--'---'--)(2n + P ,,(cos e) (17) 7. Areas of Future Work 47ThwEa 2 D"

There are numerous observations of earth-iono­ = - e) H Idr ( 2n+l )~ p ( (18) sphere cavity resonances and exi sting ionosphere and eb 47Tha Dn ae It cos source models account for most observed charac­ teristics like the resonance frequency, Q factors, with Dn = n(n + 1) - v(v + 1). Substituting (17) and (18) shape of the noise spectra, and diurnal variations of in (16) and evaluating the integrals gi ves the received power. Future work appears to be required in a systemati c 1 2ReZs (19) study of diurnal and seasonal amplitude and frequency Qi n(n + 1) ] variations of r esonances. These changes should also w/-toh [ (koa)2 + 1 be correlated with ionospheri c disturbances. The parameters of the nighttime ionosphere require also a future clarification. The variations of the ionosphere where ko= wV/-toEo. From the approximate modal parameters and the variations of the magnetic fi eld equation vector along the surface of the earth should be in­ corpor ated in future calculations [Wait, 1964b; Mad­ den and Thompson, 1964]. Much work remains to be (20) done for assessing nuclear effects on the earth iono­ sphere resonance and for di scriminating between it follows that lower level nuclear and sudden ionospheric disturb­ ances. 2 ReS ImS = ReZs . (21) w/-toh

The work of Sylvania re ported in this paper has Applying (21) and the resonance condition been supported principally by the Office of Naval Researc h under Contract Nonr 3185(00). n(n + 1) = (koa ReS)2 (22)

(19) simplifies to 8. Appendix. Q Factors 1 4 ReS ImS Qi (ReS)2+ 1 The basic definition of the Q factor of a cavity with (23) 1053

____ J ---~ . ~ ~-- -~-

which was listed as (4). When considering the dif· Galejs, 1. (1964a), Terrestrial extremely-low frequency propaga­ tion, Proc. NATO Advanced Study Institute on Natural Electro­ ferences of Er , Hr!>, Zs, and h between the day and magnetic Phenomena Below 30 kc/s, Bad Homburg, West Ger­ night hemispheres it follows that many, July-Aug. 1963, pp. 205-258 (Plenum Press, New York, N.Y.). 1 _ 2[ReS ImS!ctay + ReS ImS!nightJ Galejs,1. (June 1964b), ELF and VLF waves below an inhomogeneous Qi - (ReS!avg)2+ 1 . (24) anisotropic ionosphere, Radio Sci. 1. Res. NBS 68D, No . 6, 693-707. Galejs, J., and R. V. Row (Jan. 1964), Propagation of ELF waves The form (5) or (6) is obtained by neglecting the E below an inhomogeneous anisotropic ionosphere, IEEE Trans. integral or H integral, and by doubling the value of Ant. Prop. AP-12, No.1, 74-8:l. the H integral or E integral in (14). 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Sci. 255, 2273-2275 and 2493-2495. of the earth-ionosphere cavity modes and their relationship to Handbook of Geophysics (1960), USAF Air Research and Develop­ worldwide thunderstorm activity, 1. Geophys. Res. 67, No.2, ment Command, Airforce Cambridge Research Center (Mac- 619-625. millan Co., New York, N.Y.). . Balser, M., and C. A. Wagner (Sept. 1962b), On frequency varia­ Harris, F. B., Jr., and R. L. Tanner (1962), A method for determina­ tions of the earth-ionosphere cavity modes, J. Geophys. Res. 67, tion of lower ionosphere properties by means of field measure­ No . 10,4081-4083. ments on sferics, 1. Res. NBS 66D (Radio Prop.), No.4, 463-478. Balser, M. , and C. A. Wagner (July 1, 1963), Effect of a high-altitude Hepburn; F., and E. T. Pierce (May 9, 1953), Atmospherics with nuclear detonation on the earth.ionosphere cavity, 1. Geophys. very-low-frequency components, Nature (GB) 171,837-838. Res. 68, No. 13, 4115-4118. Jean, A. G., Jr., and A. C. Murphys, J. R. Wait, and D. F. 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Brady, A. H., D. D. Crombie, A. G. Jean, A. C. Murphy, and F. K. Koenig, H., E. Haine, and C. H. Antoniadis (Aug. 1961), Messung Stede (May 1, 1964), Long-lived effects in the D region after the von 'atmospherics' geringster frequenzen in Bonn, Z. Angew. high altitude nuclear explosion of 9 July 1962, 1. Geophys. Res. Phys. 13, No. 8, 364-367. 69, 1921. Lieberma~n, L. (Dec. 1956a), Extremely-low-frequency electro­ Brock-Nannestad, 1. (Sept. 1962), Bibliography electromagnetic magnetIc waves: I. Reception from lightning, ./. App!. Phys. 27, phenomena with special reference to ELF (1 - 3000 cis) TR 10 No. 12, 1473-1476. Saclant, ASW Research Center, La Spezia, ltalia. Liebermann, L. (Dec_ 1956b), Extremely-low-frequency electro­ Chapman, F. W., and D. L. Jones (M,!y 16, 1964a), Earth-ionosphere magnetic waves: II. Propagation properties, 1. App!. Phys. 27, cavity resonances and the propagation of extremely low frequency No . 12, 1477-1483_ radio waves. Nature 202, 654-657_ Lokken, J. 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