A theoretical model for equatorial ionospheric spread-F echoes in the HF and VHF bands

0021-9169/78/0701-0803S02.00/0

and TarcrhlPhysiu, Vol.40.~~.803to829. ~~~~.,1Q78.I4intedinNortbernWd

A theoretical model for equatorial ionospheric spread-F echoes intheHFandVHFbands HENRY G. BOOKER*

University of California, San Diego, La Jolla, CA 92093, U.S.A. and JERRY A. FERGUSON Naval Ocean Systems Center, San Diego, CA 92152, U.S.A. (Receiued 21 October 1977) Abduct-A theory of spread-F echoes is presented particularly suitable for equatorial regions, especially under presunrise conditions. In the HF band the theory is based on a spectrum of irregularities of ionization density extending from an outer scale [wavelength/(2n)] linked to the scale-height of the atmosphere down to an inner scale of the order of the ionic gyroradius. The irregularities are assumed to be aligned along the Earth’s magnetic field and to have lengths of the order of the outer scale. Inverse power-law spectral indices from 0 to 6 are considered mathematically, and from 1 to 4 numerically. Spectral indices between 1 and 3 fit observations of presumise equatorial spread-F satisfactorily. An essential feature of the fit arises from the assumed cut-off in the turbulence spectrum at the ionic gyroradius. It is also assumed, however, that a weak extension of the spectrum, probably in the form of an angular spectrum of compressional plasma waves, exists from the ionic gyroradius down to the electronic gyroradius, and on this basis a theory is presented of spread-F echoes in the VHP band. The spectrum of turbulence in the F-region plasma is pictured as maintained by travelling ionospheric disturbances caused by atmospheric gravity waves. 2. THE CHARAClT.R OF SPREAD-F

1. INlRODUcIlON

For a phenomenon

that has been known for about

half a century, it is surprising how poorly spread-F is understood. Early ionograms were made at Huancayo (Peru), which is located close to the geomagnetic equator. At Huancayo, spread-F is principally an evening and presunrise phenomenon, although it sometimes persists most of the night. An example of the presunrise phenomenon as reported by BWKER and WEUS (1938) is reproduced in Fig. 1. Ionograms of the type shown in Fig. l(a) were obtained for an hour or so before sunrise. Spread-F disappeared from the ionogram after sunrise as a result of a new F-region being created below the remnant of the F-region left over from the previous day (Fig. 1 (b), (c) and (d)). The overlap of the two types of F-region response shown in Fig. 1 (b) and (c) was explained by Booker and Wells with the aid of off-vertical transmission. It is a principal objective of this paper to understand in some detail an ionogram of the type appearing in Fig. l(a). * This author commenced this work as a Consultant to the Rand Corporation, Santa Monica, California 90406 and completed it as a Visiting Scientist at SRI Intemational. Menlo Park, California 94025 while on sabbatical leave from the University of California. 803

BOOKER and WELLS (1938) explained the ionogram in Fig. l(a) as follows. They assumed that the maximum plasma-frequency involved in the presunrise ionosphere is bound to be considerably smaller than the roughly 1OMHz that corresponds to the highest frequency from which F-region returns are visible in Fig. l(a). They therefore explained the blur as due to scattering by underdense irregularities of ionization density in the nocturnal F-region. They pictured these irregularities as being three-dimensionally isotropic with a size 4wT where T, is of the order of a few meters. Irregularities of this sixe would be small compared with the wavelength A at the low-frequency end of the ionogram and large at the high-frequency end. They showed that, in an ionized gas, Rayleigh scattering of power is independent of A instead of inversely proportional to A4, so that scattering would be roughly independent of frequency at the lower frequencies and would decrease into the noise at the higher frequencies. The explanation advanced by Booker and Wells was not readily accepted. Modem observations (M&LURE et al., 1970) con8rm the assumption of Booker and Wells concerning the low value of the plasma frequency in the presunrise F-region

804

H. C. BOOKER and J. A. FERGUSON

at the magnetic equator. But in 1938 the notion that the maximum plasma-frequency in the Fregion might be small compared with the maximum frequency for which F-region returns are obtained at vertical incidence was considered surprising. In addition, the notion that irregularities of ionization density might exist with a scale small compared with the atmospheric mean-free-path was not considered likely. The concept of irregularities of ionization density aligned along the Earth’s magnetic field had not then come into existence. Even when it did (in the 1950’s), no detailed theory of ionograms of the type shown in Fig. l(a) was immediately developed. A treatment could have been based on the theory of backscattering from amoral ionization developed by BOOKER (1956a), but in fact a theory of this type was not developed for equatorial conditions involving spread-F until twenty years later (Rros et al., 1976). The principal reason why satisfactory theories of spread-F were not quickly provided is that, observationally, the phenomenon became considerably more complex. Spread-F was frequently observed at night in many parts of the world, and it often did not take the form of an amorphous blur on an ionogram. It was quite common for the blur to have a sharp top edge near the penetration frequency and a sharp bottom edge especially at the lower frequencies, with the sharp edges seeming to coincide roughly with a reasonable position for a trace associated with coherent reflection. Qn other occasions spreading of the F-region returns seemed to consist simply of two or more traces of the type associated with coherent reflection, but slightly displaced on the ionogram with respect to each other. A panorama of situations came to light in which the spreading of F-region returns sometimes consisted simply of repeated traces, sometimes involved spreading of individual traces but with edges that had sections of sharp behavior reminiscent of a normal trace, and sometimes involved a largely amporphous blur on an ionogram. For a review of the phenomenon of spread-F see HERMAN (1966) and NEWMAN (1966). It was not even possible tosay that the spreading of F-region returns always grew directly out of the trace associated with coherent reflection. Cases OCcurred in which the spread feature first appeared separately on the ionogram at long range and then descended in range so as to attach itself to the coherent trace (AGY et aI., 1952). Furthermore, when the Jicamarca Observatory was established at a frequency of SOMHz, backscattering was frequently obtained that was strong compared with incoherent scattering, indicating irregularities in the

F-region with a scale perpendicular to the Earth’s magnetic field as small as 0.5 m but with a dimension along the field substantially greater than the north-south dimension of the array, namely 300 m (COHEN and BOWLES, 1961; FARLEY et ul., 1970). In short, the totality of phenomena associated with spread-F was such that it seemed doubtful whether a single explanation could reasonably be expected. The idea became quite firmly established, however, that field-aligned irregularities of ionization density in the F-region must be involved in the phenomenon (CALVERT and COHEN, 1961; PINEWAY and COHEN, 1961; CALVERT and SCHMID, 1964). This was verified experimentally by CLEMESHA (1964) near the magnetic equator; although the verification was challenged KINC~(1970, it was defended by KELLEHER and SKINNER(1971). Moreover, studies of the scintillation of radio-stars (HEY er al., 1946; RYE and HEWISH, 1950; Lrrrt,u and M AXWELL, 1951; HEWISH, 1952; MAXWELL and LI~~L_E, 1952; BOLTON et al., 1953; MAXWELLand DAGG, 1954; SPENCER,1955; WILD and ROBERTS, 1956; BRIOGS, 1958; KOSTEK, 1963; RUFENACH, 1971, 19721, and later of transmissions from satellites (BRIGGS and PARKIN, 1972; 1962; KOS-rER, 1966, SINOIZTON, 197Oa, b,c, 1973, 1974; AARONS et u!., 1971; FREMOUW and BATES, 1971; FREMOUW and RINO. 1973; RINO and FREMOUW, 1973; AARONS et al., 1976; BASU et al., 1976; CRANE 1976, 1977: RINO et al., 1976; RINO, 1976; RINO, 1976, 1977; SLACK, 1976; ULASZEK et al., 1976; AARON, 1977; FREMOUWet al., 1977; OTT, 1977; RAsrocr et al., 1977; UMEKI et at., 1977; WH~UEY, 197’7)

implied that the effects of irregularities in the Fregion could be experienced both as forward scattering and as backward scattering. In particular, to everyone’s surprise, scintillations in intensity were observed from time to time in communications between stationary satellites and Earth terminals at frequencies above 3 GHz (SKINNER et al., 1971; CRAFI. and WESTERLUND, 1972; TIZUR, 1973, 1974). The irregularities of ionization density involved in the scintillation phenomenon in equatorial regions are strongly field-aligned. Kosrun (1966) obtained lengths of several kilometers or more even for the narrowest irregularities that he observed (50 m). 3. m

ROLE OF ‘I’URRULENCE

It was suggested by BOOKER (1956b) that turbulence in the F-region is the cause of the phenomena of spread-F and ionospheric scintillation, and this idea was further studied by Go~rrsv~ (1959).

805

,,

..’

*

5

f IG. 3-MCORDS SUO~ING RL-‘.WL AR 4Nl.J LXFFUSE F-REGION ECHMS, UUANCAYO MAGNET/C OLWRVArORv, fEc[BRUAW IO, 1938; /A) 04b4Sm-OShOO~(L’~ OSk’%Sk5”, (Cl WhlS--d30”: (0) 0r*W-07”rSR, ?S’wEST

Fig. 1. Presunrise spread-F at the magnetic equator. Reproduced from BOOKERand WELLS(1938) by permission.

A theoretical model for equatorial ionospheric spread-F echoes in the HF and VHF bands However, at the 1959 Symposium on Fluid Mechanics in the Ionosphere (BOLIGLANO, 1959) concepts of turbulence in the ionosphere were in a rudimentary and controversial condition, and this was still true six years later (HINES, 1965). Moreover, observations of backscattering from field-aligned irregularities in the E-region associated with the equatorial electrojet (BOWLES et al., 1963) were successfully explained in terms, not of turbulence, but of a plasma instability. In the electrojet, the drift-velocity of electrons, but not of ions, can reach an acoustic velocity, and the twostream plasma instability then develops (FARJ-EY, 1963). In consequence of this success, plasma instability rather than turbulence became the preferred explanation of irregularities even at F-region levels. Moreover spread-F, both in the HF band and in the VHF band, was observed in association with artificial heating of the F-region (UTLAUT, 1970; UTL.AUT and VIOLET, 1972; CARLSON et al., 1972; MINKOFF et al., 1974; MINKOFF and KREPPEL, 1976) and an explanation of this result too has been given on the basis of plasma instability (PERKINS, 1974; CRAGIN and FIZJER, 1974). For the naturally occurring phenomena of spread-F and scintillation, many suggestions based on concepts of plasma instability have been made and pursued CLEMMOWet al., 1955; DUNGEY, 1956; MARTYN, 1959; CALVERT, 1963; LIU and YEH, 1966; REID, 1968; CUNNOLD, 1969; COLE, 1971; WONG and TAYLQR, 1971: BALSLJZY,et al., 1972; PERKINS, 1973; HUDSON et al., 1973; HMRENDEL, 1974; HUDSON and KENNEL, 1975a, b). What emerges, however, from studies of plasma instabilities turns out to be a turbulence-like spectrum of irregularities (Orr and FAFUEY, 1974; SCANNAPIECO et al., 1975; SCANPIECO and OSSASAKOW 1976; CHATUR~EDI and KAW, 1976). Moreover, in recent years turbulence-like behavior, including a turbulence-like spectrum of irregularities, has been observed in situ by satellites flying through the F-region (KELLEY and MOZER, 1972; HANSON and SANATANI, 1973; MCCLURE and HANSON,1973; DYSON et al., 1974; MCCLURE et al., 1976; PHELPS and SAGALYN, 1976; WRIGIFI. et al., 1977). Furthermore turbulence-like spectra have been obtained in observations of amplitude scintillation using both radio-stars and satellites (RUFENACH, 1971,1972; SINGLETON, 1974; TAUR, 1974; CRANE, 1976, 1977; AARONS, 1977) and in observations of phase scintillation using satellites radiating at multiple phase-locked radio frequenies (CRANE, 1976, 1977; FREMOUWet al., 1977; O-rr, 1977; UMEKI et al., 1977). The observations of phase scintillation show that the outer scale of

807

ionospheric irregularities is in excess of 1 km. By how much is not yet clear. It is not even clear how the outer scale of field-aligned irregularities should be determined (see BOOKER and MILLER, 1977). The outer scale is, however, likely to be linked to the scale-height of the atmosphere. The existence of fluctuations in the F-region with scales of the order of the scale-height of the atmosphere and longer has been known for years (BEYNON, 1948; MUNRO, 1948). They are called travelling ionospheric disturbances (TIDs) and are associated with waves in the neutral atmosphereinfrasonic waves, acoustic gravity waves and, at sufficiently low frequencies, tidal motions (Hwes, 1960). The suggestion that these atmospheric cornpressional waves might have a major role to play in connection with fluctuations of ionization density at ionospheric levels was made by HINES (1959, 1960). Suggestions concerning the relationship between TIDs and spread-F have been made by UYEDA and OGATA (1954), MCNICOL et al. (1956), MCNICOL and BOWMAN (1957), BOWMAN (1960, 1968), WHITEHEAD (1971), KOSTER and BEER (1972), BEER (1974), RC~~GER (1973, 1976) and KLOSTJXMEYER(1977). BOOKER (1975) suggested that spread-F might arise from an acoustic-type wave in the F-region with density-fluctuations comparable with the ambient density. Such a wave would ‘break,’ and strong spread-F would arise from the resulting ‘surf.’ However, Booker suggested that the compressional waves concerned might be plasma waves arriving from above the F-region whereas in fact they are more likely to be associated with compressional wave-energy arriving from below the F-region in the manner suggested by Hines. Fully developed spread-F could be associated with acoustic gravity waves that are sufficiently strong for non-linear behavior to lead to a strong spectrum of turbulence. Whatever the specific mechanism may be, there now seems to be no doubt that the phenomenon of spread-F involves a turbulence-like spectrum of irregularities, and it is appropriate to provide a detailed theory of spread-F on this basis. 4. A MODEL OF IONOSPHERICPLUCI’UATIONS It is possible to contemplate a theory of spread-F and ionospheric scintillation that involves nonisotropy at all scales, and to vary the degree of non-isotropy as a function of scale. We shall assume, however, that this is unnecessarily complicated in the present state of knowledge. Instead we shall divide the entire complex of irregularities into three regimes as follows. (a) A large-scale regime that involves scales of

H.G. BOOKER and J.A. FERGUSON

808

the order of, or greater than, the scale-height H of the atmosphere at the level under study. These irregularities might take the form of TIDs. In the horizontal direction the scale [wavelength/(2a)] may be anything from about H up to the radius of the Earth. The scale in the vertical direction is smaller than it is in the horizontal direction. (b) A medium-scale regime of irregularities extending from an outer scale TO down to an inner scale T. We picture this regime as having the characteristics of fluid-mechanical turbulence, and the motions of irregularities as involving speeds substantially less than acoustic speeds. We assume that the medium-scale regime is supplied with energy from the large-scale regime, and we take the outer scale To to be some fraction of the scale-height H. This fraction needs to be determined experimentally, but for computational purposes we shall take T, = lo-‘H.

thermic motion of the particles via wave-particle interaction. Irregularities of electron density in the small-scale regime are therefore regarded as moving with acoustic speeds. The total mean square fluctuation of ionization density due to the medium-scale and small-scale regimes of irregularities is (2, m = (AN),“+ (AN):. If we write the combined spectrum due to both regimes as S(k)m, and those due the mediumscale and small-scale regimes separately as S,(k) k (AN),’ and S,(k)(AN),’ respectively, we have S,(k)(AN),2+S,(k)(AN),2

S(k)=

-

-

(AN),‘+

(AN):

where S,(k)k

dk = 1,

S,(k)k dk = 1

(1)

The inner scale T, of the medium-scale regime will be taken to be equal to the ionic gyroradius at the level in the atmosphere under discussion. We assume that, in the medium-scale regime, irregularities of ionization density are aligned along the Earth’s magnetic field, with a length of the order of To along the field and a width across the field of k-l, running from T,, down to r. For TO-’ c k <<17;-’ we assume that the powerspectrum is proportional to kePM where pM is a spectral index characteristic of the medium-scale regime and likely to have values in the range from 1 to 4. We assume that the medium-scale regime of irregularities accounts for a contribution (AN),’ to the mean square fluctuation of ionization density N. (c) A small-scale regime of irregularities with scales transverse to the Earth’s magnetic field that run from the ionic gyroradius T, down to the electronic gyroradius T,. The small-scale regime is assumed to be two-dimensionally isotropic and to have a power-spectrum proportional to kvPS for Tie’ c k cc T,-‘, where ps is a spectral index characteristic of the small-scale regime and not necessarily identical in value to the spectral index pM for the medium-scale regime. We assume that the small-scale regime accounts for a contribution m to the mean square fluctuation of the ionization density, small compared with the contribution (AN)Mi from the medium-scale regime. The small-scale regime is pictured as having the characteristics of an angular spectrum of compressional plasma waves, as obtaining its energy from the medium-scale regime, and as losing its energy to

(3)

(4) and where in consequence S(k)k dk = 1.

(5)

If we assume that the lengths of field-aligned irregularities in the direction of the Earth’s magnetic field may be taken as infinite, reasonable expressions for the spectra of the medium-scale and small-scale regimes are given by 4wT,” = (1 + kzTo2)pJ2exp (-2kaT,‘)

(6)

4nT,f S,(k) =(1 + k2r2JpJz exp (-2k2T,“).

(7)

L(k)

The constants TM and Ts are normalizing scales chosen so as to satisfy equations (4). The factor 2 in the gaussian cut-off functions in equations (6) and (7) could be omitted, or it could be replaced by another factor such as 4. This is a matter of how the inner scales are to be defined. We have chosen the definition so that a value of T, in equation (6) equal to the ionic gyroradius gives a cut-off of spread-F in the HF band at roughly the frequency observed in practice (see Section 11). In equation (7) the part of the small-scale spectrum for which kc 17;-’ is the part that is interacting with the medium-scale regime and could be considered as part of it. Likewise, in equation (6) the part of the medium-scale spectrum for which k<< To-’ is the part that is interacting with the large-scale regime and could be considered as part of it.

A theoretical model for equatorial ionospheric spread-Fechoes in the HF and VHF bands 5. BEFIAVIOUR OF ‘ItIE SFIZI’RUM OF

where

IONOSPHERlC FLUCI’UATIONS

The

normalizing scale TM in equation (6) is such that, if in a k plane perpendicular to the direction of alignment, S,(k) were equal to S,(O)for k < TM-’ and vanished elsewhere, the correct total fluctuation of ionization density would be obtained for the medium-scale regime; similarly for the smallscale regime in equation (7). If in equations (6) and (7) the gaussian factors are omitted and the spectra are dropped abruptly to zero at k = ‘I”-’ and ‘I”-’ respectively, then equations (4) give (8)

1 - {(T 0IT)* I + 1}‘2-p~‘z {(Ps - 2)/21 Ti* Ts2 = 1 -{(TJTe)*+ l)‘*-@~* ’

For pi = ps = 2 these equations

(9)

become

TO2

(10)

TM2=ln{(T,/Ti)2+1} Ts2 =

T:

ln{(TJT,)*+

(11)

1) ’

For calculating the normalizing scales it is satisfactory to put T = 0 in equation (8) when pM ;r 3, and to put T. = 0 in equation (9) when Ts 2 3. But for pi < 3 or ps c 3 the gaussian cut-offs in equations (6) and (7) need to be taken into account. This may be done approximately by simplifying S,(k) to (4rrTM2/T,p~)k-PM exp (-2k2Ti2) when k >) To-‘, and by simplifying

S,(k)

809

(12)

y = 0.57721.

(22)

This is Euler’s constant involved in exponential integral functions. Improved accuracy in the values for TM and Ts can be obtained, if desired, with the aid of numerical integration. Equations (14) to (21) illustrate the important fact that the median scale involved in a spectrum changes markedly as the spectral index drops below 3. For spectral indices greater than 3, the median scale is of the order of the outer scale. But for a spectral index of 2, the median scale is significantly less than the outer scale, while for a spectral index of 1, the median scale is of the order of the geometric mean of the outer and inner scales. For a spectral index of ,zero, the median scale is of the order of the inner scale. The striking effect of this may be illustrated by considering only the medium-scale regime of irregularities and supposing that the spectral index pM decreases from 4 to 0 while the strength of irregularities of scale large compared with the outer scale To remains constant. the downward shift in the median scale then causes an increase in the mean square fluctuation of ionization density as pM decreases from 4 to 0. This is illustrated in Fig. 2 for To = 5 km, T = 5 m. We see that a reduction of the spectral index from 4 to 3 causes only a 3 dB increase in the mean square fluctuation of ionization density. But a reduction of pM to 2 causes a 12 dB increase, while a reduction to unity causes a 33 dB increase. If a reduction of the spectral index to zero were feasible, then an increase of 60 dB

to

(4~Ts2/Tip~)k-Ps exp (-2k2TC2)

(13)

when k >>T,-‘. In this way we obtain approximately 2”*T, )

p&f=0

(14)

(2/7r)“4(T,T#‘2,

PM=1

(19

TO

PM=~

(16)

PM23

(17)

2l’*T,,

PS’O

(18)

(2/5r)“4(T,T,)“2 ,

Ps=l

(19)

Ps =2

(20)

TM=

{21n(T,/T,)-ln2-~}1~2’ ({(PM - 2)/2)“*T, ,

T, =

T

{2ln(TJT=)-ln2-y}“*’ {(Ps - 2)/2Y’*T ,

ps 23.

(21)

SPECTRAL INDEX,B,~

Fig. 2. Decibel increase in mean square fluctuation of ionization density as pM decreases from 4 to 0 keeping the spectral density for k CCT,-* constant.

810

H.G. BANKER andJ. A. FERGUSON

would take place in the mean square fluctuation of ionization density if the spectral density for k << To-’ were kept constant. For example, let us suppose that 1% fluctuations of ionization density existed when phi = 4, so that a 40 dB increase in the mean square fluctuation would be required to cause saturation of the fluctuations. Then Fig. 2 shows that a reduction of the spectral index from 4 to I would convert the 1% fluctuations into nearly saturated fluctuations if the strength of the large-scale irregularities remains constant. We see, therefore, that equations (14)-(21) imply that a major shift in the part of the spectrum contributing to the mean square fluctuation of ionization density takes place as the spectral index decreases. The shift is from large scales to small scales and starts to become important as the spectral index decreases through about 3. The shift is major for a spectral index of 2 and enormous for a spectral index of 1. Figure 3 illustrates the spectral function S(k) described by equations (3), (6) and (7) for the following values of the parameters: (AN): = lo-*(AN)* To=5 km,

T,=Sm.

PM = 1, 2, 3, 4; Equations essentially

(23) T, = 5 cm

ps = 1, 2; 3.

mean square fluctuation of ionization density for the medium-scale and small-scale regimes is almost completely controlled by the former. In Fig 3 separate curves are shown for the medium-scale and small-scale regimes, and the addition implied in equation (3) must be performed mentally. A complete curve for S(k) therefore switches from the curve selected for the medium-scale regime to that selected for the small-scale regime as k increases through the value where the two curves intersect. Figure 3 illustrates the drop that takes place in the composite spectrum near k = T,-‘. This drop is a key feature of the spectrum and will he used to explain the high-frequency cut-off on an ionogram of the type shown in Fig. l(aj in substantially the manner proposed by BOOKER and Wm.?, (1038). The weak extension of the spectrum to values of k appreciably greater than T,-’ will be used to explain the existence of spread-F in the VHF band. From equations (6) and (7) the decibel drop near k = T,-’ in the spectrum S(k) in equation i3) is approximately the sum of

and

(24) (25)

(2) and (23) imqhat iAN),’ is not different from (AN) , so that the total

1 CM

kl

1

(27)

10 log,, {0,2/?ZG?}.

For the values of To, 7; and T, given in equation (24), the values of expression (26) are shown in Table 1 for a range of values of the spectral indices. To the entries in Table 1 must be added the number of decibels given by expression (27). We see that, if the mean square fluctuation for the small-scale regime of irregularities is more than 60 dB below that for the medium-scale regime, then a drop in the spectrum occurs near the scale T, for all situations in which the spectral indices pM and ps are less than 4. The smaller the values of the spectral indices, the greater is the drop of the spectrum near the scale equal to the ionic gyroradius. Table 1. The decibel drop in the spectrum at the ionic gyroradius when m = (Ai&’

PM P.S

Fi+.

The spectrum S of ionization fluctuations per unit (Ah') in meter2 as a function of the angular spatial frequency k in meter-‘.

0 1 2 3 4

_____

0

1

2

3

40 24 12 6 3

36 20 8 2 -1

26 10 -2 -8 -11

4 -12 -24 -30 -33

4 ____-23 -39 --5 1 -51 --60

A theoretical model for equatorial ionospheric spread-Fechoes 6. A QUALlTATlVE

THEORY OF SPRRAD-F

Quite large fluctuations ionization

density

in the

occasionally exist in the F-region (HANSON and

1973; MCCLURE et al., 1977). Associated with them, there are substantial variations in the maximum ionization density in the F-region, and consequently in the penetration frequency of the F-region. Under such conditions there is a range of frequencies for which patches exist in the F-region that are reflecting, and holes exist that are transparent. The severest forms of spread-F are probably associated with such conditions. For simplicity, however, let us consider more moderate conditions for which the fluctuations of ionization density are less drastic. Table 2 shows the values of certain quantities of interest (a) in connection with VHF radars of the type existing at Jicamarca, and (b) in connection with ionosondes. For back-scattering, the observing systems are sensitive to scales in the direction of the line of sight equal to 1/(4~) times the radio wavelength A. We see that, for Jicamarca, the relevant scale is 0.5 m, while for an ionosonde, it runs from about 1 m up to about 25 m. At F-region levels in equatorial regions the ionic gyroradius is about 5 m, and it therefore occurs in the range explored by an ionosonde. The wavelength for which h/(4~) is equal to the ionic gyroradius corresponds to a frequency of about 5 MHz, and this is comparable to the penetration frequency of the F-region. The confusion caused by this rough coincidence has probably contributed substantially to the delay in developing a detailed understanding of spread-F because, under many circumstances, it results in the cut-off frequency for backscattering being of the same order of magnitude as the cut-off frequency for coherent reflection. Table 2 also shows the Fresnel scale defined as SANATANI,

TF = (~Az/.R)“~

(2% where z is an F-region height taken to be 400 km. At the low-frequency end of an ionogram the Fresnel scale is comparable with a likely value for the outer scale of field-aligned irregularities in the ionosphere. But at the high-frequency end of an ionogram, or at the Jicamarca frequency, the FresTable 2. Response scales and Fresnel scaIe~ jicamarca f h hl(4?r) TF

SOMHZ 6m 0.5 m 1.2 km

Ionosonde 25-l MHz 12-300 m l-25 m 1.7-8.7 km

in the HF and VHF bands

811

nel scale is likely to be less than the outer scale T,. Nevertheless the Fresnel scale is large compared with the inner scale Ti, assumed to be equal to the ionic gyroradius. Thus the spectrum of irregularities likely to exist in the ionosphere runs from scales above the Fresnel scale to scales well below it, and this has important consequences. Scattering is caused by irregularities having scales both above and below the Fresnel scale listed in Table 2. But scales above TF cause refractive scattering and those below TF cause diiiractive scattering. Both refractive and diffractive scattering are important for transionospheric propagation, and are the cause of the phenomenon of scintillation. However, for a radar operating at a frequency sticiently high to avoid coherent reflection from the ionosphere, refractive scattering is irrelevant, and diffractive scattering is only important for backscattering at scales in vicinity of A/(4r) listed in Table 2. The situation is quite different, however, for a radar that is operating at a frequency sufficiently low to experience coherent reflection from the ionosphere. Scintillation then comes fully into play even for a radar. Below the penetration frequency of the ionosphere, irregularities with scales larger than TF listed in Table 2 cause refractive scattering, leading to wandering of the direction of arrival of the wave at the receiver. Scales larger than TF can even lead to more than one ‘normal’ onto the ionosphere from an ionosonde, and consequently to multiple traces on an ionogram (UYEDA and OGATA, 1954; MCNICOL et al., 1956; MCNICOL and BOWMAN, 1957; BOWMAN, 1960, 1968; MATHEWSand HARPER, 1972). On the other hand, scales that are less than the Fresnel scale TFIbut that are nevertheless appreciably greater than the scale A/(47~) listed in Table 2, lead to diffractive scattering but not to backscattering. Such scattering, unless strictly in the forward direction, causes energy to arrive back at the ionosonde at a time different from that for the coherent reflection, thereby spreading the trace on the ionogram (CALVJZRT and COHEN, 1961; CALVERT and SCHMID,1964). Just below the penetration frequency, where the delay-time for strictly vertical propagation is large, the effect of difiractive scattering in a generally forward direction is to reduce the delay-time, thereby spreading the trace downwards and leaving a sharp top edge. But at lower frequencies the increased pathlength caused by diffractive scattering in a generally forward direction spreads the trace upwards and leaves a sharp bottom edge.

H. G. BANKER and J. A. FERGUSON

812

Let us now suppose that, in addition to the larger scales, scales of order A/(4n) and less come into play. These cause diffractive scattering in all directions including the backward direction. Backward scattering then occurs for the upgoing wave before it is reflected, and for the downcoming wave after it is reflected. If sufficiently strong, this backscattering further spreads the trace, and can do so in such a way as to blur the sharp edges. For frequencies below the penetration frequency of the ionosphere the effect of ionospheric irregularities on ionograms is therefore as summarized in Table 3, classified by spatial frequency. There are almost always fluctuations in the ionosphere with scales of the order of the scale-height H or more, and these cause some wandering of the direction of arrival at the ionosonde. If extension of the spectrum downwards to the Fresnel scale TF becomes important, multiple ‘normals’ onto the ionsophere from the ionosonde may occur, causing multiple traces on the ionogram. If the active spectrum becomes extended below the Fresnel scale but not down to scales of the order of hl(4~), each trace spreads, but a sharp top edge is preserved near the penetration frequency and a sharp bottom edge at lower frequencies. Further extension of the significant spectrum of irregularities down to scales of the order of A/(4~r) causes each trace to spread further in such a way as to smear the sharp edges. It is reasonably clear that this constitutes a rough description of the development of spread-F as frequently observed with ionosondes. Let us now consider frequencies above the penetration frequency of the ionosphere. As the frequency increases through the penetration frequency, the role of refractive scattering and of diffractive scattering by irregularities having scales appreciably greater than hl(47r) becomes restricted Table 3. Effect of ionospheric irregularities on ionograms below the penetration frequency, classified by spatial frequency kl(2ri Active spatial frequencies only sufficiently small k k
all frequencies except possibly k B h/A

Associated phenomenon wandering of direction of arrival same, but multiple traces may also occur spreading, downwards near penetration frequency, upwards at lower frequencies further spreading, tending to smear the sharp edges

to scintillation phenomena associated with tram,ionospheric propagation. On an ionogram, onl) backscattering from irregularities having scales of the order of h/(4rr) is important for frequencies appreciably above the penetration frequency. However, consideration must be given to the likely existence of a cut-off in the spectrum of the medium-scale regime of irregularities at the ionic gyroradius 7: (- 5 m) as illustrated in FIN. 3. The fact that the range of values for the scale A/(47r) for an ionosonde extends (see Table 2) $a smaller scales than the inner scale T, for the medium-scale regime of irregularities means that the highest frequencies on an ionogram are exploring scales in the ionosphere less than T,. It is only for the lower frequencies on an ionogram, therefore, that one can expect to obtain backscattering from the ionosphere easily. But at these frequencies backscattering is usually confused by coherent reflection and by forward scattering, both refractive and diffractive. There are two ways of avoiding this confusion. One is to use sufficiently high frequencies, and to off-set the reduction in ionospheric backscattering at k = T,-’ by an increase in the power and the antenna size of the radar. Using VHF, this solution is available at the Jicamarca Observatory. An alternative procedure, however, is to retain the use of HF frequencies but to exploit situations in which the penetration frequency of the ionosphere is ex.~ ceptionally low-less than, say, 2 MHz. To adopt the latter procedure one must examine ionograms taken either near the F-region trough at about L = 4 or ones taken in the period before sunrise in equatorial regions. The latter possibility is the simpler because it avoids aurora1 complications and avoids the consequences of a steeply sloping magnetic field. Presunrise conditions at the magnetic equator are the very conditions under which the ionogram shown in Fig. l(a) was made. At the magnetic equator the horizontal component of the electric field in the ionosphere usually reverses in the evening (FARLEY er al., 1970; WOODMAN, 1970), after which time it transports ionization existing in the F-region downwards to levels where it disappears by recombination. Before the end of the night, this process has frequently reduced the maximum ionization density in the F-region to quite low values (MCCLURE et ul., 1970). The maximum ionization density in the presunrise equatorial F-region is often comparable with the value that one would deduce by extrapolating the plasmaspheric Led profile down to F-region levels. The latter value is about 5 x

A theoretical model for equatorial ionospheric spread-F echoes in the HF and VHF bands 1010m-3 if deduced from the ‘typical’ plasmaspheric profile of CHAPPW et al. (1970). In Fig. l(a) the penetration frequency of the F-region should be read as about 1 MHz. Virtually the entire ionogram is therefore concerned with backscattering from the F-region unconfused by coherent reflection and by forward scattering. In Fig. l(a) the backscattering begins to weaken as the frequency increases above about 5 MHz, and it disappears into the noise at about 1OMHz. We proceed to interpret this in detail as due to a decrease in the spectrum of field-aligned irregularities of ionization density at a scale of the order of the ionic gyroradius as illustrated in Fig. 3. 7. THEORY OF BACXSCAFIELD-ALIGNED IRREG-

G FROM LONG

A detailed theory of spread-F should be simplest for presunrise conditions at the equator such as those illustrated in Fig. l(a). At frequencies appreciably above the penetration frequency (say greater than 2MHz in Fig. l(a)), refraction of the rays joining the ionosonde to a scattering irregularity is small, and we shall assume that it may be neglected. Let us assume initially that field-aligned irregularities may be taken not only as straight but also as of infinite length. Scattering is then a twodimensional phenomenon in a plane perpendicular to the direction of alignment; along the direction of alignment coherence is retained. At the magnetic equator this means that the directions of arrival of backscattered waves at an ionosonde are spread in a magnetically east-west plane, but not out of this plane. The effect of taking the length of the fieldaligned irregularities to be finite instead of infinite will be discussed in the following section. In spite of the long history of the theory of radio backscattering from atmospheric irregularities using the Born approximation (BOOKER and GORDON, 1950; BOOKER, 1956a; BOOKER, 1959; SALPETER and TREIMAN, 1964; RIOS et al., 1976), a theory capable of dealing with field-aligned irregularities that behave coherently in the direction of alignment does not appear to exist in the literature, but is required in order to study spread-F. In the scattering medium let (x, y, z) denote Cartesian coordinates, for which the x axis is along the direction of alignment. Let N denote the mean ionization density in the vicinity of the origin, and let AN(y, z) denote the excess ionization density at the point (x, y, z) due to the field-aligned irregularities. Neglecting the effect of the Earth’s magnetic field in the calculation of the capacitivity E, the mean square fractional deviation of capacitivity at

wavelength

813

A is

where r, is the classical radius of the electron (2.82~ lo-” m). In equation (29) A should, strictly speaking, be the wavelength taking into account the mean ionization density N but, under circumstances such as those illustrated in Fig. l(a), we shall assume that A may be taken as the wavelength in free space. In discussing the details of the scattering process, we shall assume that N, m and (Ae/c)* are independent of position, but subsequently we shall assume that they vary slowly from point to point in the ionosphere. Let us assume that the transmitter is located in the (y, z) plane at a distance rl’ from the origin, and let us assume initially that it is linearly polarized with its electric field parallel to the direction of alignment. Let the power radiated by the transmitter be P and let us assume for the time being that this includes the power-gain of the transmitting antenna in the direction of the origin. At the point (x, 0,O) let the distance from the transmitter be RI’, so that 2

R*0={(r*o)z+x2}1’2~r,O+L

2r10’

(30)

Let the distance from the transmitter to the point (0, y, z) be rl, and let that to the point (x, y, z) be R1. Then (31) Let us assume initially that the receiver is not necessarily at the same position as the transmitter, but that it is located in the (y, z) plane. Let rzo be the distance of the receiver from the origin and let r, be its distance from the point (x, 0, O), while RzO and R2 are the distances of the receiver from the points (0, y, z) and (x, y, z) respectively. Then

Rzo%r2’+$ 2

and R,kr,+$j.

(33) 2

The complex electric field produced by the transmitter at the distant point (x, y, z) is approximately l’* exp (i(ot - k”R,)} 0

r1

(34)

H. G.

814

BOOKER and

where f‘ is the impedance of the medium (approximately 377 ohms), w is the angular frequency of the transmitter and k0=2n/A. In equation (34) the denominator should read I?, but it has been rc placed by r,“. The direction of the electric field E,, is not quite that of the x axis, but we shall neglect the small discrepancy. The extra electric moment per unit volume created by the incident wave at the point (x, y. z) is E,,As and, using the Born approximation, the complex electric field at the receiver is approximately

x exp(-ik”&) dxdydz 0 r?

(35)

where R, in the denominator of the integrand has been replaced by rzo. Because we are assuming that A&/e is not a function of x, the integration in equation (35) with respect to x may be separated from the integrations with respect to y and z, and then performed. Substituting into equation (35) from equations (30~(33), we obtain approximately

-~

JJ w

X

exp l-jk”x*/(2r”))

dx

J. A. FERCWWN then becomes &P‘ 1,2( k0)3/2(+))‘/ E = _ i 2n. ) @af”z X

exp [j{,t - k”( r,’ + r2’) - $ n}] r,0r20

F(k, .--k,) (41)

where F(k) is the ho-d~e~ional Fourier transform of A&/e, which is a function of r = (y? 2). It follows from equation (41) that the mean power density at the receiver is

From Parseval’s theorem IF(k)/’ may be related to the autoco~elation function of be/~. If S(k) is the Fourier transform of this autocorrelation function, then ~Fw=

SW

Jm J=(Ade)2dy .-m .-.X

143)

dz.

Let A be an area of the (y, z) plane large compared with the transverse scales of the fieId-aligned irregularities but small compared with the distances to the transmitter and receiver. Then equation (43) may be written (44)

IF@)/‘= ~~)~AE/&)~A

c-2

2 exp {-jk’(r, -zC -@z E

+ rJ) dy dz

(36)

where

and equation

(42) becomes S@, -k@j2

1 1 yj=y+-y. r

r1

1

A.

(45)

(37) By writing this result in the form

f.2

Evaluating the integral with respect to x, we obtain for the complex received field

s

X

JJ -

A&

- exp (-jk’(r, -m -ez E

+ r2)) d y dz.

(38)

The double integral with respect to y and z may now be converted into a Fourier integral by writing approximately k”(r, - rlO)= k10 . r

(39)

I k”(r2- rzo) = kzo . r

(40)

where r is the two-dimensional vector (y, z), k,” is a vector of length k” directed from the transmitter to the origin (the direction of incidence), and kzO is a vector of length k” directed from the origin to the receiver (the direction of scattering). Equation (38)

it may be interpreted in words as follows. The received power, per unit plane angle in the plane through the ~ansmitter and receiver ~~ndicul~ to the direction of alignment, per unit incident power density, per unit area in the plane perpendicular to the direction of alignment, is (47) Equation (47) gives the scattering coefficient for infinite straight field-aligned i~~~arities. If the transmitter is linearly polarized, not parallel to the direction of alignment, but perpendicular to it, then the expression on the right hand side of equation (47) should be multiplied by the square of the cosine of the angle between the direction of the

A theoretical model for equatorial ionosphericspread-Fechoes in the HF and VHF bands

electric field at the transmitter and that at the receiver. When the receiver is at the same location as the transmitter we have, in accordance with equation (37), r,O= rzo= 2r0 1 k2 = -k, = (2m/h)f

(48)

(49)

where & is a unit vector directed from the radar to the scattering location. Equation (47) then gives the back-scattering coefficient uB

=f;(yS(p)

(50)

appropriate for long straight field-aligned irregularities. In terms of the mean square fluctuation of ionization density this becomes, in accordance with equation (29), a, = ~~,‘A(AN)‘S{(~~T/A)&

(51)

For long straight field-aligned irregularities that are two-dimensionally isotropic we drop the unit vector f in equation (51), thereby obtaining UB = $r,*A(AN)*S(4r/A)

(52)

and the function S(k) may then be identified with that appearing in equation (3). 8. FINITJC LENGTH OF FIELD-ALIGNED IRREGULARRIW

In the previous section we have assumed that field-aligned irregularities are infinitely long. This means that the Fresnel scale TF defined in equation (28) is small compared with the length of the irregularities. The model of ionospheric irregularities described in Section 4 assumes that the length of field-aligned irregularities is of the order of magnitude of the outer scale T, for the mediumscale regime of irregularities. Moreover, in accordance with equation (l), a reasonable value for T, at a height of 400 km is 4 km. Using Table 2, we see that the condition T,
(53)

815

and this evaluates to (2r0A)“*Fr{T~(2r0A)“*}

(54)

where Fr(u) is the Fresnel integral Fr(u) =

b

“exp {-j(d2)u2}

du.

(55)

In the previous section the value of To in expression (54) was taken as infinite, so that the value of this expression was then (2r”A)“* Fr(m).

(56)

By dividing expression (54) by expression (56) we see that the correction factor required in equations (38) and (41) to allow for the finite length of field-aligned irregularities is

Fr(m)

.

(57)

When the transmitter and receiver are at the same location, the correction factor for fieldaligned irregularities at perpendicular distance r from the common location becomes, in accordance with equations (48), Fr{Tol(rA)“*} Fr(m)

(58)

’

This correction factor applies to the received complex electric field. For the received power the correction factor for finite length of the irregularities is Fr{ To/(rA)“*} * Fr(m)

*

(59)

This is the factor that must now be included in equations (50~(52). A plot of IFr(u)lFr(m)r vs u is shown in Fig. 4.

Fig. 4. Illustratingreplacement of the oscillatory Fresnel factor (expression (59)) by the smoothed Fresnel factor (expression (61)).

816

H. G. Boorosa and J.

The oscillation about the value unity in Fig. 4 when u > 1 is associated with the assumption that all field-aligned irregularities have the length of the local outer scale To of the medium-scale regime of irregularities. If we assume that there is enough spread in the lengths of irregularities to smooth out this oscillation, the full-line curve in Fig. 4 can be replaced by a curve such as the broken one, for which the functional dependence on u is 2uZ (1 f

(2U2)2}3/2

(60)

Use of this smoothed function converts the correction factor (59) into 2T02/(rA)

[1+{2T,‘/(rA)}‘]“’ This is the coefficient o, in order to perpendicular the order of

(w

factor by which the backscattering in equation (52) must be multiplied permit field-aligned irregularities at distance r to have finite lengths of To.

A. FERGUSON

to an isotropic antenna, be G(B)---the same for transmission and for reception. For continuous wave transmission, the power received by backscattering from infinitely long field-aligned irregularities intercepting the element of area dS is

where CT, is given by equation slowly varying function of r and irregularities of finite length To correction factor (61), thereby sion (63) into -- P

A2

2 To21(rA.)

(52) and is now a 0. For field-aligned we incorporate the converting expres-

oB (r. 0)

4ar* 4~ [1+~2~~/(rA)~~]~‘~

-

G2( 0) dS.

r

(64) For pulse-transmission this expression is only valid while the pulse of short duration r is crossing dS. Moreover, the backscattering from dS contributes to the power received after a delay-time I where $ct=r.

9. CALCULAllON OF EQUATOIUAL BA-WG FROM FIELD-ALIGNED IRREG-

We are now in a position to calculate, for a radar located at the magnetic equator, the backscattered power received from field-aligned irregularities in circumstances when intervening refraction may be neglected. We shall assume that the irregularities are straight and horizontal. Let the magnetically east-west plane through the radar be the (y, t) plane, with the z axis in the vertical direction and the y axis in the westerly direction so that the x axis is in the direction of the earth’s magnetic field. We now assume that the various parameters controlling the backscattering coefficient vB [namely,W%*, (WSZ, PM, Ps,To, '& and T.] are functions of position in the (y, z)

plane. These functions are supposed to be slowly varying in comparison with the transverse scales of the field-aligned irregularities causing backscattering. Let dS be a macrosopic element of area of the (y, z) plane at perpendicular distance r from the radar, and let the direction of the radius from the radar to dS make an 8 with the upward vertical, so that dS=dydz=rdrdB.

(62)

Let P now be the actual power radiated by the antenna, and let the power-gain of the antenna in the (y, I) plane in the direction 8, measured relative

(65)

Let PT be the peak power, and let the power in the pulse as a function of time f be (66)

8(t)TPr

where 6(t) is the unit impulse function. Then the power received at delay-time t from field-aligned irregularities intercepting the element of area dS is C7PT A2 2T021(rA) -4nr2 47r [l +{27’~/(rA)}2]1’2

oB(r, 0 G2(e)i3(r-act) r

dS.

(67)

Replacing dS by r dr dt? and integrating with respect to r from 0 to 03 we obtain for the power received at delay-time t between the angles 0 and 0 + de to the vertical in the magnetically east-west plane c7Pq-A2

2T02/(+ctA)

2 10 (47r)* [1+{2T0*/(;ctA) ]

a,(h

(ict)2

e)

Q(e)

de.

(68) The total power received delay-time t is therefore c~P,h=

pR = m

from all directions

at

J

n/2 2T,2/($ctA) 1 (tCt)2 -_& [1+ (2 To21(~ctA)}2]“2

x a&t,

e)Gz(e) de.

(69)

In this result the factor arising from finite length

A theoretical model for equatorial ionospheric spread-Fechoes in the HF and VHF bands of field-aligned irregularities may be taken out of the integral if the outer scale T, is assumed to be independent of position (y, L) in the magnetically east-west plane. If however T, is a slowly varying function of y and z, it must be interpreted in equation (69) to mean T,[y, z] = T,[$ct sin 0, ict cos 01.

(70)

The same interpretation applies to the T, implicit in uB as a result of the spectrum-factor S appearing in equation (52). In this factor the scales Ti and T, should likewise be interpreted as Ti[y, z] = T&ct sin 8, fct cm 01 I T,[y, Z] = T&t

sin 0, fct cm 01.

(71) (72)

In addition, the mean square deviation of electron density m appearing in equation (52) is in general a slowly varying function of y and z, and needs to be written (A#[y,

z]= (AN)‘[1cct sin 8, $ct cos e].

(73)

If desired, the spectral indices can also be regarded as slowly varying functions of y and z, so that phl(y, z)=p,($t sin e,$crcos e) I p&y, z)=p&ct

sin e,;ctws e).

(74) (75)

With these precautions we may substitute for uB from equation (52) into equation (69) to obtain for the received power 2T,‘/($ca) [ 1+ (2 T,‘/(i c~A)}‘]“~

(76) When suitable functions are inserted into equation (76) for the power polar diagram G(B) of the antenna, for the normalized ionization fluctuationspectrum S(k) and for the functions appearing in equations (70)-(75), we obtain the ratio of the received power to the transmitted power as a function of the radio wavelength A and the delay-time t. Th$ is the type of information that is displayed on an ionogram such as that shown in Fig. l(a). At the Jicamarca wavelength it is what is displayed on an A scope under conditions of spread-F when the direction of the beam is adjusted to be at right angles to the earth’s magnetic field. 10.APPLXATlON

OF THJ!. THEORY

For Jicamarca we can assume a uniformly illuminated square antenna of side w both in the east-

817

west direction and in the north-south direction (w = 300 m). The appropriate expression for G(8) in equation (76) is therefore

WI=?

4mu2 sin (m/h)

sin 8

(7rw/h) sin e

2 1 ’

(77)

For a modern ionosonde the transmitted power Pr is roughly independent of wavelength but, as the wavelength is reduced, the antenna has increased power-gain in the vertical direction thereby giving, for coherent reflection, a roughly uniform response over the HF band in the absence of absorption.The antenna pattern of an ionosonde is complicated, but it may be roughly represented analytically as a vertically pointing end-tie antenna of length 1 (-20m), so that

1

~0s ((27rllA sin’ (e/2)} *. 1+(41/A)sin2(8/2)

(78)

However, for the original automatic ionosonde installed at Huancayo, records from which are reproduced in Fig. 1, the antenna pattern was that of a horizontal dipole above ground. For a northsouth dipole, G(8) would be approximately proportional to cos’ 0 and for an east-west dipole to coos40. In fact the dipoles were erected at an angle of 45” to the magnetic meridian so that, for coherent reflection, they responded equally well to the ordinary and extraordinary magnetoionic components, which are linearly polarized at the magnetic equator. The power polar diagram of the antenna in a plane perpendicular to the magnetic meridian therefore had a behaviour intermediate between cos’ 8 and cos4 0. Since the precise effect of the terrain is uncertain we simply take G(0) to be proportional to ~0s’ 8. Moreover, the power-gain of the antenna did not change radically with frequency because a succession of dipoles was used, switching taking place at the gaps appearing in the records shown in Fig. 1. It was therefore necessary to compensate for the lack of power-gain at the shorter wavelengths by using more radiated power. To produce, for coherent reflection, a roughly uniform response over the HF band in the absence of absorption, it is necessary for PT to be roughly proportioned to A-‘. For comparison with the ionograms reproduced in Fig. 1 it is therefore appropriate to substitute into equation (76) P,G2(8) = Po(A,/A)’ cos6 0

(79)

where P,, is the product of the peak power radiated at a reference wavelength A0 and the square of the small power-gain of the antenna arrangement in the vertical direction.

818

H. G. BOOKERand J. A. FERGUSON

For the dependence of m on y and z appearing in equation (73) we may make the rough approximations that {(AN)‘}“’ is proportional to the ambient ionization density N and that this is a function of the height z only. We then have (BN)2=m(N(z)l“

(80)

where m is assumed to be independent of position in the ionosphere. Equation (76) for the received power then becomes

Table

4. Evening

profile log,,,N,, = 8.7, 6370 km

it

1

2,x

300

A n 1. n 1130

2 350 - l/200

l/30

B,

Table

5. Presunrise

de.

(81)

profile log,,N,= 6370 km

Figure 5 shows the profiles of ionization density that we shall use for evening and presunrise conditions in equatorial regions. They are specified analytically by the method of BOOKER (1977) using the parametric values listed in Tables 4 and 5. Well above the level of maximum ionization density in the F-region the profiles fit in with the ‘typical’ Lm4 plasmaspheric profile of CHAPPELL et al. (1970). The maximum ionization density for presunrise conditions correponds to a plasma frequency of 1.2MHz (cf. Fig. l(a)). 11. RESULTS OF CALCULATIONS (IONOGRAMS)

To produce a theoretical ionogram for comparison with Fig. l(a), we use in equation (81) the profile specified by Table 5. In addition, we substitute for P,G’(B) the value given in equation (79), and use for S(k) the spectrum described in Sections 4 and

8.7,

2,) = R, =

1

300

,?I

A,-,, n

-0.4?43/(+R,! l/300

n

x{N($ct cos S)}%j~~GZ(~)

I,, = K,; -=

-0.43431~R,i

l/30 l/30

B,

5. This spectrum is illustrated in Fig. 3 for a particular height. We shall assume that the spectral indices pM and ps are independent of position. We shall do calculations for pw = 1. 2, 3, 4 but keep ps constant at the value 2. It is also necessary to specify the functions appearing in equations (70), (71) and (72). We shall assume that Ti and T, are independent of position and have values 5 m and 5 cm respectively. But. in accordance with equation (l), the value of the outer scale To depends on the scale-height of the atmosphere, and therefore on height. We take T,=a+pz

(82)

and put (Y= 0,

p = 1o-2.

(83)

When all these substitutions are made into equation (81) and the angular integral is evaluated numerically, we obtain the received power PR as a function of the wavelength A (frequency f, and the delay time t (range ict). Contour maps for this received power are shown in Fig. 6. To the decibel markings on the contours should be added the negative number of decibels given by 10 log,, (AN/NY depending on the choice made for the fractional fluctuation of ionization density. The decibel contour-markings adjusted in this way then give the number of decibels by which the received power exceeds 10 log,,{r,2h~‘PocT/(32aZ)).

Fig. 5. Profiles of ambient ionization density.

(85)

At ranges in excess of 300 km in the contour maps shown in Fig. 6 there is a frequency-band

A theoretical model for equatorial ionospheric spread-Fechoes in the HF and VHF bands

600

FREQUENCY

IN MHz

Fig. 6. Contour maps of received power in a diagram of virtual height versus frequency under presunrise conditions at the magnetic equator. To the contour markings in decibels add the negative values given by expressions (84) and (85). where the contours are closer together than elsewhere. For the lower values of the spectral index pi, this band constitutes a ‘cliff’ separating a ‘high plateau’ towards the lower end of the HF

band from a ‘low plateau’ in the VLF band. As pM increases from 1 to 4 the steepness of the clii decreases. The cl8 behaviour is caused by the presumed cut-off in the medium-scale regime of irregularities at a scale of the order of the ionic gyroradius. It is reasonably clear that the high plateau region of the contour maps in Fig. 6 is potentially capable of explaining the spread-F ap-

pearing in Fig. l(a). The low plateau region of the contour maps is also potentially capable of explaining spread-F as observed at Jicamarca when the necessary adjustments for power and antenna are made. The edge of the blur on the ionogram in Fig. l(a) probably corresponds to a signal-to-noise ratio of unity. Let us therefore derive contour maps of the signal-to-noise power ratio. The signal-power has been calculated in Fig. 6 and we now consider what the noise-power is likely to have been. Determination of the wavelength dependence of

H. G. BANKER and J. A. FERGUSON

820

noise in the HF band is normally difhcult. However, the conditions under which the ionogram shown in Fig. l(a) was taken were particularly simple. The record was made in 1938 in the Andes Mountains of Peru at a time and place for which man-made interference was very low. This is confirmed by the relative lack of vertical interferencelines on the records in Fig. 1. Moreover the ionogram in Fig. l(a) was made when the penetration frequency of the ionosphere was only about 1 MHz, so that propagation of atmospherics from distance sources of thunderstorms was unimportant for most of the record. This is confirmed by the lack of background darkening of the record except at the very lowest frequencies. The observations were made under presunrise conditions when local thunderstorms are unlikely to have existed. Solar noise is irrelevant because the ionogram was made before the sun rose. It seems likely therefore that, for most of the ionogram in Fig. l(a), the relevant background noise is galactic. Let us, therefore, assume that the background noise-power PN appropriate to the ionogram shown in Fig. l(a) is such that (International Radio Consultative Committee, 1964) PN = a(hlh,J2 3

036)

where a is a constant and ho is a reference wavelength. On this basis the contour maps of received power shown in Fig. 6 become the contour maps of received signal-to-noise power-ratio presented in Fig. 7. To the decibel markings on the contours in Fig. 7 should be added the negative number of decibels given by expression (84). The decibel contour-markings adjusted in this way then give the number of decibels by which the received signal-to-noise power-ratio exceeds 10 loglo{r,‘A~.‘Poc~/(3282a)}.

(87)

Examination of the contour maps in Fig. 7 shows that, for ranges in excess of 300 km, the high plateau of signal-to-noise ratio towards the lower end of the HF band is relatively uniform for pM in the range from 1 to 3. This plateau is separated from the low plateau in the VHF band by a cliff that becomes less steep as pM is increased from 1 to 3. For pM = 4 the high plateau has considerable downward slope towards high frequencies, and the cliff separating it from the low plateau is less pronounced. If values of pM greater than 4 were used, the contour maps would slope downwards more-orless steadily from low to high frequencies and would not bear much resemblance to the ionogram reproduced in Fig. l(a). Reasonable fit between the

theory and the observations is given by values of pW in the range from 1 to 3, with phi = 2 being perhaps the best. In comparing the theoretical ionograms in Figs, 6 and 7 with the observed ionogram in Fig. l(a) it should be noted that the scales are different. It should particularly be realized that the frequency scale in the observed ionogram, although nonlinear, is not logarithmic. Although not reproduced, we have redrawn the theoretical ionograms in Fig. 7 so as to have the same format and size as the observed ionogram in Fig. l(a). The visual similarity of the observations to the theory is thereby improved, especially for pM = 2. It is, of course, unsatisfactory to have to compare theoretical contour maps of the type drawn in Figs. 6 and 7 with a mere blur on an ionogram. However, in spite of the roughly. forty years that have elapsed since the ionograms in Fig. 1 were made, no observed contour maps have been constructed that could be compared in a fully quantitative manner with the theoretical contour maps presented in this section. Likewise, there seems to be no quantitative radio data from which one can reliably estimate the value of (AN/N)’ required to explain backscattering of the type illustrated in Fig. l(a). Even for the relatively sophisticated observations of spread-F by CLEME~HA (1964) at 18MHz and by KELLEHER and SKINNER (1971) at 27.8 MHz using steerable antennae, no data were reported from which the ionospheric backscattering coefficient could be estimated numerically. Clemesha did, however, suggest that the field-aligned irregularities of ionization density are ribbons rather than rods. Such a possibility is incorporated in the theory presented in Section 7; it is merely a matter of using equation (51) rather than equation (52). It may be noted that the ionogram reproduced in Fig. l(a) suggests the existence of repeated scattering from the F-region with intermediate reflection from the ground. It may be inferred, therefore, that the scattering must have been remarkably strong. Moreover, the strong scattering was occurring in circumstances when the ambient ionization density in the F-region was remarkably low. The percentage fluctuation in ionization density must, therefore, have been quite high, possibly approaching 100%. The in situ experiments of HANSON and SANATANI(1973) and of MCCLURE er al. C1977) confirm that fluctuations of ionization density close to 100% do actually occur in the equatorial Fregion. The theory presented in this paper for presunrise

A theoretical model for equatorial ionospheric spread-Fechoes in the HF and VHF bands

821

600

600

FREQUENCY

IN MHz

Fig. 7. Contour maps of received signal-to-noise power-ratio in a diagram of virtual height versus frequency under presunrise conditions at the magnetic equator. To the contour markings in decibels add the negative values given by expressions (84) and (87).

spread-F at the magnetic equator in the HF band is sufhciently successful to warrant application of the same model of ionospheric fluctuations to the more complicated spread-F situations that involve refraction in a major way (see Table 3), and also to the phenomemon of ionospheric scintillation encountered in satellite communications in the VHF, UHF and SHF bands. The former has not yet been attempted but results of the latter wlll be presented in a separate paper (BOOKER and MILLER, 1977).

12.RFsuJm OF cALcuLA TloNs (JICAMARCA)

Let us now examine the low plateau in the contour maps of Fig 6. Here backscattering is in the VHF band, and takes place from the small-scale regime of irregularities for which we have arbitrarily selected a value 2 for the spectral index p* Jn this band of frequencies interest attaches to the Jicamarca Observatory for which f = 50 MHZ, cr = 15 km, and the antenna polar diagram is given by equation (77). Using these values in equation (81),

822

H. G. BANKERand J. A. FERGUSON

RECEIVED

POWER !N DECIBELS

ABC”E

TR~ISMITTED

POWER

Fig. 8 Height depend&x of received power for prestmrise unctions at the Jicamarca Observatory for spread-F (continuous curves) and incoherent scattering (broken curve).

we obtain the variation of PRIPT with $ct shown in Figs. 8 and 9. Figure 8 employs the presunrise profile of ionization density shown in Fig. 5, and Fig. 9 employs the evening profile. The broken curves in Figs. 8 and 9 correspond to incoherent scattering for which the backscattering coefficient is frezN(z) per unit votume. The received power for incoherent backscattering is

JJ 42

0

m of ionization density as a ratio to the local value of v, and also for various values of the ratio (A&‘/p appropriate to the smah-scale regime of irregularities alone. It is to the latter markings that attention should fn-st be directed, because spread-F at Jicamarca depends on the small-scale regime. The strongest returns received at Jicamarca under evening conditions of spread-F are about 50dB above those for incoherent scattering (WOODMAN and LA Hoz, 1976). This corresponds roughly to the curve marked

7r

N&t cos 6)G2(@, 4) sin 0 d6 d$

(AN),Z/N2 = 1O--’

(88)

-a

in Fig. 9. This estimate for the strength of the small-scale regime of irregularities depends on the assumption that the outer scale for this regime is the ionic gyroradius and the inner is the electronic gyroradius. It also depends on the choice of the value 2 for the spectral index pS of the small-scale regime. If the value 3 were chosen for ps, one would obtain roughly

where G(@, &)=,,

47rw’ sin{(aw/h) sin 6 cosd} c (n-~/h) sin 8 cos # xsin{(7rwlh) sin 8 sin (6) 2 (7rw/h) sin B sin fj

I ’

(89)

The curves presented in Figs. 8 and 9 are marked for various values of the mean square fluctuation

(AIu),2fI@ = lo-‘.

EVENING

RECffVED

(90)

POWER IN DEClBELS

ABOVE TRANSMITTED

POWER

Fig. 9. Height dependence of received power for evening conditions at the Jicamarca Observatory for spread-F (continuous curves) and incoherent scattering (broken curve).

(91)

A theoretical model for equatorial ionospheric spread-Fechoes in the HF and VHF bands The strength of the small-scale regime of irregularities varies from values such as those estimated in equations (90) and (91) down to values small enough for incoherent scattering to dominate. This is at least 5 powers of ten less than the values quoted in equations (90) and (91) and probably 6 powers of ten less. The weakness of the small-scale regime of irregularities contrasts strikingly with the strength of the medium-scale regime. If we regard the situation depicted in Fig. l(a) as illustrative of strong spreadF and take note of the multiple blurs appearing on the ionogram, one can estimate very roughly that (AN),‘/N’

= 10-r

(92)

= 1.

(93)

or possible even that (AN’)d/N

It was by comparing the values in equations (90) and (91) with those in equations (92) and (93) that the value lo-’ used in equation (23) emerged. This value could, of course, easily need correction by a power of ten either way. It was by using the figure of lo-’ for the ratio of the strength of small-scale regime of field-aligned irregularities to that of the medium-scale regime that the values of m/N’ were marked on the curves in Figs. 8 and 9. The difficulty of assessing the relative strengths of the medium-scale and small-scale regimes of irregularities, even to the nearest power of ten, reveals an obvious need for coordinated quantitative experimental studies of the backscattering of radio waves from irregularities in the F-region as a function of frequency in the HF and VHF bands treated as a single band. 13. DISCUSSION

It is reasonably clear that spread-F, at least as encountered under presunrise conditions at the magnetic equator, is explicable in terms of long field-aligned irregularities with a spectrum of the type illustrated in Fig. 3. There is a medium-scale regime of irregularities for which the scales transverse to the direction of alignment exceed the ionic gyroradius, and there is also a weak extention of this spectrum to scales less than the ionic gyroradius. The spectral index for the mediumscale regime is likely to be in the range from 1 to 3. Under presumise conditions at the magnetic equator scales greater than about 25 meters are not directly used for generating returns appearing on ionograms (see Table 2). Even so, the spectrum at the large scales is important because it has a major

823

influence on the strength of the spectrum at the smaller scales that directly cause backscattering. Moreover, when one comes to study the aspects of spread-F that involve refraction in a major way (see Table 3), the larger scales come directly into play. They cause roughly forward scattering of both the diffractive and refractive types, and these scattered waves are themselves reflected from the Fregion except at frequencies appreciably greater than the penetration frequency of the ionosphere (CALVERT and COHEN, 196 1; CALVERT and S-ID, 1964). The drop in the spectra of Fig. 3 that takes place at a scale of the order of the ionic gyroradius is an essential feature of the behaviour of spread-F. It is what prevents the blur on ionograms from extending up to the high-frequency end of the records. Moreover, in the HF band, there are no reports of the motions of irregularities involving random velocities of acoustic magnitude whereas, in the VHF band, random velocities approaching acoustic magnitude do appear (BALSLEY et al., 1972; WOODMAN and LA Hoz, 1976). This suggests that the medium-scale regime of irregularities in Fig. 3 is associated primarily with mechanical turbulence, whereas the small-scale regime may arise from an angular spectrum of compressional plasma waves. Some care may be necessary in using VHF observations as a diagnostic tool for the main spread-F phenomenon in the HF band and for scintillation phenomena at VHF and higher frequencies. So far as velocities are concerned, the small-scale regime of irregularities involved in VHF backscattering will certainly participate in the motions of the medium-scale regime of irregularities. But the same may not be entirely true for the strength of the irregularities. Only a very small fraction of the total mean square fluctuation of ionization density is associated with the small-scale regime. There could easily exist important variations in the strength of the small-scale regime, both in time and in space, without there being much corresponding variation in the vastly greater strength of the medium-scale regime. We clearly need a theory of how the entire spectrum of irregularities in the F-region comes into existence. It needs to be a theory that explains why, apart from regions complicated by the aurora, the spectrum of irregularities is strong near the magnetic equator. It needs to be a theory that not only explains why spread-F occurs, but also explains why, in most places at most times, spread-F does not occur. It needs to be a theory that explains for the daytime equatorial F-region why, although

824

H. G. BANKERand J. A. FERGUSON

the ambient ionization-density is high, nevertheless the fluctuations of ionization-density are so low that spread-F is rarely experienced (CALVERT and SCHMID, 1972). The theory not only needs to explain why equatorial spread-F occurs primarily at night, but also why it is particularly associated with two periods during the night (BOOKER and WELLS, 1938; LYON et al., 1964; FARL.EY et al., 1970; M&LURE et al. 1970; RASTOGI, 1977). One is the presunrise period, and the other is a period during the evening when the F-region is raised significantly by action of an electric field (WOODMAN, 1970). It is not yet certain that the irregularities of ionization density involved in spread-F are initiated by a plasma instability. It is not even certain that the driving phenomenon occurs primarily in the

1968; TIZSTUDand VASSEUR, 1969; STERLING et al., 1971). They are also observed in situ (NEWTON et al., 1969; POTTER et al., 1976; TKINKS and MAYR, 1976). Acoustic gravity waves in the neutral atmosphere affect the plasma via ion drag. But the ion drag also limits the growth with height in the strength of the fractional fluctuations of neutral density (GERSHMAN and GRIGOR’YEV, 1965; HINES, 1968b; KLOSTERMEYER, 1969a, b; LIU and YEH. 1969; TEZSTUD and

VASSEUR, 1969;

Hoo~a,

t970:

CLARK et al., 1971; TE~TUD and FRAN~XXS, 1971; KLOSTERMEYER, 1972a, b, c; LIU and KL,OSTERMEYER, 1975; YEH et al., 1975; KLOS~EKMEYERet

al., 1975). This limitation to growth is reduced near the magnetic equator where the particle motion for acoustic gravity waves arriving from higher ionized atmosphere. The best known fluctuations of latitudes can be largely parallel to the Earth’s magnetic field. Moreover, because of the lower ionization-density in the F-region are the travelling ionospheric disturbances (BEYNON, 1948; MUNRO, ionization density, the limitation on upward growth 1948, 1950, 1953, 1958; BEYNON and THOMAS, caused by ion drag is less by night than it is by day, and it is particularly low under presunrise condi1954; UYEDA and OGATA, 1954; PRICE, 1955; tions in equatorial regions. It is also low underTOMAN, 1955; MUNRO and HEISLER, 1956a, b; neath an evening equatorial F-region that has been MCNICOL et al., 1956; MCNICOL and BOWMAN, significantly raised by action of an electric field, 1957; HEISLER, 1958; BIBL and RAWER, 1959; RAO and RAO, 1959; THOMAS, 1959; BOWMAN, thereby leaving relatively un-ionized gas below. It is possible, therefore, that spread-F, at least in its 1960, 1968; HLTNSUCKERand TVETEN, 1967; equatorial form, arises at those times and places GEORGES, 1967, 1968a, b; TITHERIDGE, 1968, where there is a relative lack of ion drag. This 1971; FIALER, 1970; GOODMAN,1971; REDDI and RAO, 1971; HERRON and DONN, 1973; NAGPAL et could permit the fractional fluctuation of neutral density in an acoustic gravity wave to be amplified al., 1973; SEW et al., 1973; EVANS 1975; with height enough to permit the fluid velocity to TOMAN, 1976). TIDs are driven by acoustic gravity exceed the phase velocity, at least in the frequency waves in the neutral atmosphere (HINES, 1959, 1960, 1962; PFEFFER, 1962; PRESS and HAR- band where the phase velocity is low. Non-linear KRIDER, 1962; P-WAY and HINES, 1965; FRIED- action in the neutral atmosphere then causes the wave to ‘break,’ There is overturning of the neutral MAN, 1966; MIDGLEY and LIEMOHN, 1966; HOOKE, 1968, 1970; YEH and LIU, 1974; FRANCIS, atmosphere and updwelling of regions of compara1975; MAYR and VOLLAND, 1976; WEINSTOCK, tively low ionization density. Spread-F in its fullest development is an aspect of the resulting ‘surf.’ 1976; WEINSXJCK and HYDE, 1976; SMITH, 1977). The idea that strong spread-F occurs when and While such waves are weak in the lower atmoswhere acoustic gravity waves in the neutral atmosphere, the fractional fluctuation of neutral density phere are able to break should be compared with has to grow with height at the rate of half a neper the observations described by KEI.LEY. et al., per scale-height in order to keep the vertical com(1976) but contrasted with their interpretation. The ponent of energy-flow uniform in the absence of interpretation that we have made in terms of absorption. Even after allowance has been made acoustic gravity waves places the primary instabilfor absorption due to atomic and moiecular viscosity and thermal conduction (Prrr~wau and HINES, ity, not in rhe ionospheric plasma, but in the neutral atmosphere. Nevertheless plasma physics 1963; HINES, 1968a; VOLLAND, 1969a, b) and due still plays an important role. Not only is it involved to ion drag, the waves are frequently strong enough in ion drag but, when once irregularities of density in the F-region to generate visual ripples on ionograms (WRIGHT and KNECHT, 1957; AGY et al., in the plasma have been caused via ion drag, nonlinear behaviour of the plasma then creates a 1959). By means of incoherent scattering, these turbulence-like spectrum of field-aligned irwaves are detected above the level of maximum regularities of ionization density. This part of the ionization density in the F-region (THOME, 1964,

A theoretical model for equatorial ionospheric spread-Fechoes in the HF and VHF bands calculation can probably be adapted from the corresponding theory for a disturbance Sated by a plasma instability (SCANNAPIECOand OSSAKOW, 1976). Moreover, plasma physics is certainly involved in the small-scale regime of field-aligned irregularities of ionization density that are likely to be an angular spectrum of compressional plasma waves. But it is not certain that these waves arise from a plasma instability created by wave-particle interaction. Landau damping is likely to be the dissipative process for such an angular spectrum of compressional plasma waves. But an obvious source of energy for creating the waves is the energy cascading down the medium-scale regime of irregularities that is probably associated with dynamical turbulence. We need a theory describing how some of

J., WHITNEYH. E. and ALLEN R. S. AARONSJ.

AARONS

J., MULLENJ., Wrurtm~ H., ~~TTN E., BHAVNANIK. and WHELANL.

825

this cascading power is converted into compressional plasma waves at a scale of the order of the ionic gyroradius and then converted at smaller scales into the thermic motion of particles. Acknowledgements-This work was supported in part by the Defense Nuclear Agency under Contract DNAOl-76c-0410. One author (H. G. B.) wishes to thank WALTERG. CHESNUTof SRI International for discussions concerning the structure of field-aligned irregularities in the ionosphere. Thanks are also due to CHARLES L. RINO and ROBERT C. LMNGS~~N for providing access to unpublished results of the DNA Wideband satellite experiment. The final choices adopted in this paper for the values of a number of the ionospheric parameters were made with the results of the DNA Wideband satellite experiment in mind. Thanks are due also to THOMAS A. CROFT for discussions concerning travelling ionospheric disturbances and acoustic gravity waves.

1971 1977

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NEWTONG. P., F%LZ D. T. and VOLL.ANDH. OTT E. and FARLEYD. T. Ch-r R. H. PERKINSF. W. PERKINSF. W. -R R. L. WHELPSA. D. R. and SAGALYNR. C. DAY M. L. V. and COHEN R. Prrmv~y M. L. V. and HINTS C. 0. PlTIEWAY M. L. V. and HINES C. 0. PO~ITERW. E., KAYSER D. C. and MAUSBERGERK. PRass F. and HARKRIDERD. PUCE R. E.

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