The theory of the reflection of low frequency radio waves in the ionosphere near critical coupling conditions

Journal of Atmospheric and Terreetrial Physica,1973,Vol.35, pp.51-62. Pergamon Pretrs.Printedin NorthernIreland

The theory of the reflection of low frequency radio waves in the ionosphere near critical ooupbg conditions M. S. t!kTH Cavendish Laboratory,

Cambridge, England

(Received 23 June

1972)

Ah~~e~-The theory of the reflection of radio waves vertically incident on the ionosphere is studied for conditions where the coupling between the ordinary and extraordinary waves is strong. Only smoothly varying electron distribution functions N(z) are used, so that reflections at discontinuities of N or of its gradient are avoided. There are then two main reflections, one from near the level where X = 1 f Y and one from near where X = 1. There is a critical coupling frequency which depends on the magnetic dip angle. When the frequency or the dip angle is continuously changed so that this critical condition is passed, the polar&cation of the X = 1 + Y reflection changes over from that of an ordinary to that of an extraordinary wave, though the range of frequency in which this change occurs is found to be much greater than was previously predicted. One hind of observed reflection from the level near where X = 1 has been called a ‘coupling echo’ which would be strongest for frequencies near the critical coupling frequency. It is shown that such an echo would not occur with a smooth electron distribution function, so that it must be explained by some discontinuity in the ionosphere.

1. I~TR~DU~TI~~ MANY workers have observed the reflection from the ionosphere, at vertical incidence, of radio waves in the low frequency range (50-500 kHz). Most (WATTS and BROWN, 1954; KELSO et al., 1951) have used the ionosonde technique with pulse modulation. They have shown that the waves returned from the ionosphere can be separated into several discrete reflections, each with a charac~ristic equivalent height of reflection. Two of these reflections have been explained as coming from the levels in the ionosphere near where X = 1 + Y and X = 1. (X and Y are the standard symbols of magneto-ionic theory (RATCLIFE‘E, 1959) and are defined in Section 2.) It is convenient, for brevity, to refer to the observed reflections, or echoes, as ‘the X = 1 + Y reflection’ and ‘the X = 1 reflection’, respectively. In this paper we shall be ~terested in the amplitudes and polarisations of these reflections for various frequencies and latitudes. For a particular ionospheric model and latitude, there is a certain frequency, f,, called the ‘critical coupling frequency’ (BUDDEN, 1961, p. 413) which usually falls within the low frequency range. When the frequency of observation, f, is varied so that it increases from f -c f, to f > fc, magneto-ionic theory predicts a marked change in ~larisation of the X = 1 + Y reflection (BUDDEN, 1961, Section 19.10). Alternatively this change may occur at a constant frequency if some ionospheric parameter changes with time, so as to make f, change through the critical coupling condition f = f,. When the X = 1 + Y reflection and the X = I reflection are separated in time, as with the ionosonde technique, it is nearly always observed that the X = I + Y reflection is much the stronger. A possible explanation is that each wave suffers absorption at the level where it is reflected, but the collision frequency is much greater at the lower level where X = 1, so that the reflection from that level is much 61

52

M. S. SMITH

weaker. This would apply if the X = 1 reflection is simply the reflection of the ordinary wave at the level X = 1 where its refractive index would be zero if collisions were absent. Certain reflected echoes have been observed at low frequency from near the level X = 1, and are called ‘coupling’ echoes (KELSO et al., 1951; LINDQUIST, 1953). They occur when f is near to fc and it has been suggested that they arise in a different way, namely by conversion of an upgoing extraordinary (or ordinary) wave to a downgoing ordinary (or extraordinary) wave near the level X = 1 when f is near to fc. When the ionospheric reflection coefficient matrix R is calculated for a single frequency, it contains information about all the separately reflected waves. When f is not near fc the only appreciable reflected wave comes from near the level where X = 1 + Y, and its polarisation is independent of the polarisation of the incident wave. Then the reflection coefficient matrix R is singular (Section 4). For frequencies near to fc, however, the presence of a coupling echo would make R non-singular because the reflected wave would then contain two independently reflected components. It is shown in Sections 5 and 6 that for an ionosphere with a smoothly varying electron distribution function N(z), the matrix R remains singular for a wide range of frequency including fc. It is concluded that the ‘coupling echo’ would not occur with a smooth N(z). This agrees with the conclusions of BUDDEN (1972). One method of studying the X = 1 and X = 1 + Y reflections would be to calculate the reflection coefficient matrix R at several frequencies and then to perform a Fourier synthesis to give the response to a pulse. The two reflections would then be separated in time. Work on this method is in progress but for the anisotropic ionosphere it is complicated and beyond the scope of the present paper. Here the X = 1 + Y reflection is studied by using a model of the ionosphere for which the X = 1 reflection can be shown by other methods (Section 5) to be very weak. The method of calculating R and the ionospheric model used are given in Section 3. Section 4 discusses the concept of a ‘simple reflection’, that is an ionospheric reflection with a singular reflection coefficient matrix. The polarisation of the X = 1 + Y reflection and the ratio of extraordinary to ordinary wave as conditions change from f > f, to f -=c fc are studied in Section 5, and the ‘coupling echo’ is considered in Section 6. 2. NOTATION AND DEFINITIONS A Cartesian coordinate system 2, y, z is used with the z axis vertically upwards and the x and y axes chosen so that the Earth’s magnetic field is in the Z-Z plane. This paper is concerned only with plane waves whose wave-normals are vertical. It is assumed that the ionospheric plasma is cold and that its constitutive relations are influenced by electrons only. The principal symbols used are as follows: admittance matrix A Cartesian components of the electric field of the wave E,, E,, E, frequency of observation f critical coupling frequency electron gyrofrequency (taken equal to 1.2 &IHz) plasma frequency Cartesian components of the magnetic field of the wave

53

The theory of the reflection of low frequency radio waves refractive index for ordinary and extraordinary electron concentration reflection coefficient matrix

wave respectively

~~ V>?rf

&Y sinW/cos 0 characteristic admittance of free space angle between the Earth’s magnetic field and the vertical 0 such that fc =f effective collision frequency for electrons wave polarisation, E,/E, value of p for the ordinary and extraordinary wave, respectively.

0, V P PO?Pe

For a given height x the admittance matrix R are defined by:

matrix

A

and the reflection

coeflkient

(1) R = (1 - A) (1 + A)-l

(2)

1964, Ch. P., with C = 1 for vertical incidence). In particular for the free space below the ionosphere let the horizontal components of the electric field of the upgoing and downgoing waves be respectively Eni, Evi and EE,,, E,,. Then (BUDDEN,

(3) Equation (2) can also be used within the ionosphere to give the reflection coefficient that would be observed for the region above the chosen level, if there were a sharp boundary at tbis level, with free space below (BUDDEN, 1964, Ch. 4). 3. METHOD The admittance

matrix

;;=(A

A

for vertical incidence

;j+(;r

;)A-A(;

satisfies the differential equation ~)-A(~I”)A

(4)

(BUDDEN, 1964, equation (4.45); the Tij are defined by equation (4.33) of this reference). This was solved numerically, for a particular f and 0, by a stepwise integration Each integration was started at a proceeding downwards through the ionosphere. great height where there can be only upgoing waves. It can be shown that the starting value of A is then given by

A=

(% -

’

1 - po2, Po2%) p&no -

~&b

ne), n&l

- nA - po2) .

(5)

54

M. S. SMITE

After the intuition, the value of A at the bottom of the ionosphere was used in (2) to give the reflection coefficient matrix. This paper is concerned with understanding the mechanism of the reflection processes and the changes of polarisation of the waves. It is not concerned with any particular model of the actual ionosphere. Some of the reflections studied below may be weak and it is therefore important to avoid conditions which could give unwanted weak reflections. Such reflections could occur at discontinuities of N(z) or of dN(z)/d.z (BUDDEN and COOPER, 1962), or of deiV(z)/dza (HAYES, 1972). In particular they could appear erroneously in the calculated reflection coefficient, if dN/& or d2N/dzz is appreciable at either of the levels where the integration is started or stopped. To avoid this it was assumed that X (proportional to N(x)) is X = X,/(1 + eez)

(6) (see Fig. 1). Then for large positive z where the integration is started, X has an approximately constant value close to X,, and its first two derivatives are negligible. Similarly for large negative z where the integration is stopped, X is close to zero and again its derivatives are negligible. Typical values used were c( = O-2 km-i, with integration over the range z = + 30 km to - 40 km. In the calculations 2 (proportional to electron collision frequency) was assumed to be given by 2 = 2, e-@# where 2, is a constant. Typically j3 was O-1 km-l.

Fig. 1. The function X(z), propo~ion~ to the electron concentr&tion N(z), for s typical oaaestudied in this paper. In this example the frequencyf is 160 kHz so that the free spece ws;vtilengthis 2 km, and a in (0) is 0.2 km-l. The levels whereX = 1 and X = 1 + Y Bse marked for Y = 8 aorrespondingto an electron gyro fiequenoyf= = 1.2OMHz.

(7)

The themy of the reflection of low frequency m3i~ wttv~~

55

4. THY,CONCEPT OB& ‘SIMPLEREFLECTEON’ When studyiug the reflection of eontinoue wakes from the ionosphere at vertical incidence it is useful to resolve the incident wave, after it has entered the ionosphere into the two components ordinary and extraordinary. These are reffected at different levels. Sometimes only one of these two reflected waves is observed, for example because one wave penetrates the ionosphere, or because one is much weaker than the other. Suppose that only the extraordinary wave is reflected. When it leaves the ionsophere its polarisation, pl has a value which is uuaffeoted by the state of polarisation of the incident wave. Then px, the polarisation returned from a linearly polarised incident wave with E, = 6, and pz, the polarisation returned from a linearly polarised incident wave with E, = 0, are equal, so that p = p1 = pz, where whence

PI = -R,,IRm

PZ = -RdRm

det (R) = 0

(8) (91

(BUDDEN,1961, p. 500). .A reflection of this kind will be called a ‘simple reflection’. When the extraordinary wave starts downwards from its reflection level, near where X = 1 + Y, its polarisation is at first that of a pure extraordinary wave, p&). As it travels down, however, it can generate some dowugoing ordinary wave, with poIa~~tion p,(z), by a coupling process. The composite wave therefore has a polarisation p which is neither pt nor pO. The coupling process does not affect the relations (8), (9) for a simple reflection. For a simple reflection from an isotropic ionosphere it has been shown by BUDDEN and COOPER(1962) that the equivalent height of reflection is:

where the reflection coefficient R is a scalar. This result may abo be apphed to an anisotropic ionosphere where R is now any Rfj. The derivative aR/8f can be calculated as AR/Aft i.e. from the values of R at two adjacent frequencies. The three independent R+j (RI, = R2,) give three nearly equal values for A”, If there are two reflections present, for example the X = 1 and X = 1 + Y reflections, then the polarisations p1 and p2 are not in general equal; the two alternative linearly polarised incident waves,, imphed in the definition (3), one with Eai = 0 and the other with E,, = 0, contain different relative amounts of ordinary and extraordinary wave, so that the resultant polarisations of the reflected waves are different. In this ease det (R) # 0. For au example of this see Fig. 7. 5. THE X = I + Y REFLECTION Experimental observations at vertical incidence are usually made at a fixed latitude, and thus 0 is constant. Then either the frequency is varied (CARLSON,1960) to pass through the critical coupling conditions f = $,, or the frequency is kept constant (Kxnso et G&, 1951) over a sufficiently long time for the ionosphere to change through critical coupling conditions. For a particular ionospheric model, it is simplest, when studying the theory, to vary either frequency (at constant 0) or 0 (at constant frequency).

56

M. S. SMITH

In this section, the values X, = 24,Z, = O-1atf = 150 kHz, (X, CCl/j”, 2, K l/j’ for other frequencies) were used (See Fig. 1). The method of Section 3 computes the overall reflection coefficient and includes both the X = 1 + Y and the X = 1 reflections. We wish to study the polarisation of the X = 1 + Y reflection. This is possible for this model, at low frequency (75-300 kHz), because the X = 1 reflection is very weak. This was confirmed in the three following ways : (a) If the X = 1 contribution is very weak, the overall reflection consists predominantly of the reflection from X = 1 + Y which is a simple reflection (Section 4). This can be tested using the criterion (9) which would give p1 = pa; this was indeed found to be the case. (b) The special case 0 = 90” (Earth’s magnetic field horizontal) is useful because the ordinary and extraordinary waves are then propagated independently. In (4) T,, and T,, are then zero, and A and R are diagonal. Then R,, is the reflection coefficient for the ordinary wave (the X = 1 reflection) and R,, is the reflection coe~cient for the extraordinary wave (the X = 1 -+ Y reflection). Results for this case are given in Fig. 2, which shows that the X = 1 + Y reflection is much stronger than the X = 1 reflection at these frequencies, for the model ionosphere assumed here.

lo3 ,;

150

300

Frequency, kHz

Fig. 2. Amplitudes of the X = 1 (ordinary) and X = I + Y (extraor~n~y) reflections versus frequency, when the Earth’s magnetic field is horizontal, 0 = 90’. In this special case the two waves are independently propagated and therefore untiected by coupling. The curves show that the ordinary wave reflection is much weaker. (c) The equivalent height of reflection h’ was calculated from (10). It was found to be (i) the same for &, R,, and Rz2,(ii) very nearly independent of 0, and (iii) slightly greater than the height where X = 1 + Y. The reflection coefficient and the polarisation, p, calculated are therefore those for the X = 1 + Y reflection. Figure 3 shows the value of p (plotted in the complex p plane) calculated for various 0 at frequencies 75,100,150 and 300 kHz. The continuous curves show how

The theory of the reflection of low frequency radio waves

57

Fig. 3. The complex plane showing the polarisationp of the X = 1 + Y reflection for various values of the frequency, f, and the angle 0 of the Earth’s magnetic field. For this example the model of the ionosphereused was that of equation (6) and the collision frequency was given by equation (7).

p varies when 0 is varied and the frequency f is kept constant. The broken curves show how p varies when f is varied and 0 is kept constant. It is important to find the relative amounts of ordinary and extraordinary components in the emergent wave. It is shown in the Appendix that if the (complex) amplitudes J’,, and Fe respectively of these components are measured by the major axes of their polarisation ellipses, then WIi6

= (P -

pJl(l

-

PPJ.

(11)

The value of p0 is given (RATCLIFFE, 1959) by: iZ, PO= 1 -

X(x) -

iZ(z) -

il+ (

8, ( l-X-iZ

=* 11

(12)

and it depends on height, z, for a particularf and 0. At very low z, for example at the ground, X M 0 and Z >> Z,, so that p. +a - i, and is independent off and 0. If this value for p0 is used to calculate Fe/F0 for the polarisations of Fig. 3, the results shown in Fig. 4 are obtained. In Fig. 4 the expected transition from predominantly ordinary to predominantly extraordinary as 0 increases through 0, (or f increases through fJ is not clear. The curve for 75 kHz does not cross the line IFJFJ = 1 at all, and although the 150 and 300 kHz curves do cross it when 0 is near 0, the proportion of extraordinary at greater 0 does not keep increasing. The use of p,, = - i is, however, misleading for the following reason.

M.

s. sEI[lTn

E’ig.4. The relative amplitudes and the phw differencefor the component extraordinary and ordinary waves in the X = f + Y reflection when it emerges from the bottom of the ionosphere. Fe/F0is given by equation (11). For these curves the value pO= -i wag used.

The phenomenon of ‘limiting polarisation’ (BUDDEN, 1961, p. 432) affects the downcoming wave, especially at the lowest frequencies, for which Y?, the coupling parameter (FGRSTERLINO,1942), is largest. Limiting polarisation is a coupling process tending to maintain a constant value of p as z decreases, and below a certain level (the bottom of the ‘limiting region’) p does not change with z. The value of p. which ought to be used is therefore the limiting polarisation, pL, of a wave which, just above the limiting region, is a pure downgoing ordinary wave. The theory of limiting polarisation (HAYES, 1971) shows that pL is approximately equal to p0for a real level in the ionosphere; for this particular model, and these frequencies, this level is found to be z = -25 km. When the value of pa appropriate to this level is used in (11) instead of p0 = -i, the values of Pb/F’o shown in Fig. 5 are obtained. Now the curve for 75 kHz cxosses the line f_lt’B/PJ= 1 near where 0 = O,, and all three curves show a monotonic increase in the p~po~ion of extraordinary wave with increasing 0 or f. The range of frequency, at fixed 0, in which the polarisation changes from predominantly ordinary to predominantly extraordinary is much larger than was predicted by BUDDEN(1961, p. 431). It is possible, however, that other electron distribution functions N(z), for example with a smalIer gradient dN/dz near where X = 1, would give a smaller range for the transition. In experimental work (CARLSON,1960) it has been observed that the sense of rotation of the ~larisatio~ ellipse reverses when the frequency increases, and this would occur when Imfp) changes sign. A more detailed experimental study of the

The theory of the reflection of low frequency radio waves

89

Fig. 5. Similar to Fig. 4 but the value of p. was given by equation (12) with X and 2 calculated from (6), (7) tit the level where z = -25 km.

shape of the polar&&ion ellipse, such as would be obtained by rne~~g of p (complex), has not been reported as far as the author is aware.

the value

6. TIIE ‘CO~?LIXU ECHO’

KELSO et al. (1951) and L~NDQUIST (1953) predicted that the coupling echo would

be present near criticrtl unction because these are the conditions where coupling between the ordinary and extr~ord~&ry waves is strongest. The strong coupling between downgoing extraordinary and downgoing ordinary nesr the level where X = 1, produces the ~pproxim&~ly equal extraordinary and ordinary components in the wave reflected from near X = 1 + Y, when f fi: fC. A coupling echo could be produced if there were also strong coupling between upgoing extraordinary and do~goiug ordinary, and between upgoing ordinary and downgoing extraordinary, and would otiginrtte near where X = 1. The presence of a coupling echo in addition to the X = 1 + Y retie&ion would make A # pz (BUDDEN, 1961,p. 500). I&ure 6 shows the calcul&ed values of pr and pz for the case f = 150 kEz in Fig. 3, for values of 0 very close to 0,. The values of p1and pz are very nearly equal, and remain so right through oritical coupling, so that the reflection remains ‘simple’. If there were e couphng echo whose amplitude is greatest when 0 is near O,, then the values of p1 snd pe would separate nesr critical coupling conditions. The fact that Pl = pa very close to critical con~tious means that the ‘ooupling echo’ does not exist with any detectable amplitude.

M. S. SMITH

21.00

t -I.

0

1 - 8

1 - .s

RetpI

20&l 20.25” @ 20,0”@ 19 .?5”@ 19.5”@

--0.1

l9@

Fig. 6. The complex p plane showing values of p1 (marked as crosses) andpz (marked &8 circles) for the overall ionospheric reflection (for a frequency of 150 kHz) very near critical coupling conditions. The values of 0 for each pl, pz pair are shown on the diagram. 0, is 20.0’.

The results in Fig. 6 should be contrasted with Fig. 7 which is a similar diagram but for a case where there is an appreciable ordinary wave reflection from near where X = 1. It is seen that pi and pz are now appreciably different for all values of 0 and not only when 0 M 0,. The values of p1 and pz are closest together when 0 is small, that is when the ordinary reflection is weakest. Im

(p) 0 30”

042”

I x 200

,

X30” -I

0

0100

@W

5”O

-150

Re(p)

1

x5*

X42’

-1 I

Fig. 7. Similar to Fig. 6 but for a model with a constant value of 2 ( = O-05) for a = 1 + Y and theX = 1 reflections frequency of 300 kHZ. 0, = loo. Both theX are appreciable and the figure shows that p1 # pz.

The theory

of the reflection of low frequency radio waves

61

BUDDEN (1961, p. 418, and 1972) has demonstrated the independence of the upgoing and downgoing waves near an isolated coupling point in the complex X plane; strong coupling between upgoing and downgoing waves near an isolated coupling point should therefore not occur, and ‘coupling echoes’ should not be present. As the ordinary wave reflection near X = 1 is generally very weak at these frequencies, the observed X = 1 reflections called ‘coupling echoes’ must have another explanation. A possible explanation is reflection at a discontinuity AN in the electron distribution N(z) (GNANALINGAM and WEEXES,1955; GARDNERand PAWSEY,1953; BELRO~Eand BURKE, 1964): the amplitude of the reflection is determined by the ratio An,%,, (n is the refractive index) which would be greatest near the level X = 1 where n is small for the ordinary wave. Reflections due to this mechanism would not appear for the smoothly varying model assumed here. 7. CoNCLUsIoNs The effect of the transition through critical coupling on the reflecting properties of the ionosphere at low frequencies has been studied, and two main conclusions can be made: (i) The expected change in polarisation of the X = I + Y reflection occurs, but the range of frequency in which the change over takes place is much greater than was previously thought. (ii) The ‘coupling echo’ does not exist when the electron distribution function is smoothly varying. author is indebted to Dr. K. G. BUDDEN, F.R.S., for his advice and (also for the derivation in the Appendix), to Mr. J. A. RATCLIXFE,F.R.S., for helpful comments on the text, to Prof. 3-I.V. WILKES, F.R.S., for permission to use the TITAN computer, and to Mr. XI. G. W. HAYES for calculating the limiting polarisations used in Section 5. He has been in receipt of a grant from the Science Research Council during the period of this research.

Acknowledgements-The

encouragement

REFERENCES BELROSE J. S. and BURKE M. J. BUDDEN K. G.

1964 1961

BUDDEN K. G.

1964

K. G.

1972 1962 1942 1953 1955

BUDDEN

K. G. and COOPER E. A. F~RSTER~I~~ K. GARDNER F. F. and PAWSEY J. L. GNANALIN~AM S. and WEEKES K.

BUDDEN

HAYES M. G.

W.

KELSO J.M., NEARHOOBH. J., NERTNEY R. J. and WAYNICK A. H.

LINDQGIST R. RATCLI~E J. A. WATTS

J.

M.

J. geophya. Res. 69, 2799.

Radio Waves in the Ionosphere. Cambridge University Press. Lectures on Magnetoionic Theory. Blackie, London. J. atmos. terr. Phye. 34, 1909.

J. atmos. teTr.Phys. 24, 609. ~oc~f~~. Tech. E~ect~oakust59,lO. J. atmos. terr. Phys. 8, 321. Physics of the Ionosphere, p. 63. Phys. Sot., Lond. Proc. R. Sot.

1971 1951

A324, 369. Annls Gkophys. ‘4, 215

1953 1959

J. atmos. terr. Phys. 4, 10, Ma~eto~on~c Theory. Cambridge Univer-

1958

J. geophys. Rea. 63, 717.

sity

Press

M. S. SMITH

62

J. geophys. Res. 59, ‘71.

1954

WATTS J. M. and BROWN J. N.

Reference & also made to the following unpublished material : CARLSON H. N.

1960

HAYES M. G. W.

1972

Scientific reports 129, 139. Penn. State Univ. U.S.A. Ph.D. thesis, Cambridge University.

APPENDIX Let the wave at the bottom of the ionosphere have polarisation p, where (1)

P = J%I%

For a pure ordinary wave, E,/E, = pO, and for a pure extraordinary wave, E,/E, = pe = l/p0 (RATCLIFFE, 1959, equation (2.5.9)). In the free space below the ionosphere, for a,pure ordinary wave let E, = EaOeiot. The energy flux II, in the wave is proportional to: 1%12 + lE,12 =

l&l2 (1 +

(2)

pope*).

Similarly for a pure extraordinary wave the energy flux II, is proportional to:

IJM’ (1 +

VP,P~*)

l-%,/~ol~ (1 + pope*).

=

(3)

We take the ratio of the amplitudes extraordinary/ordinary to be the square root of the ratio of these energy fluxes, so that (4) (H$-10)1’2 = I%e/PJ%oI. Now for the composite wave -% = -%, + % Ey = PJL

+ %IP,

(5)

= PK.

Then E

,E

Y

=

p

=

PO +

z

tEdEd/,-%

(6)

1 + P,,/Ez,)

so that %e/~dL

=

(P -

p,)/(l - PP&

(7)

Hence (4) is

I(P - P,)/U - PPob For the phases we use the F&sterling variables (BUDDEN, 1961, p. 397) F,, = (pfly - E&p,2 - 1)1’2

Fe = (p$,

- Ey)(p02 -

1)1’2.

(9)

The phase difference is, with the use of (6): arg E6 - mg E6 = mg (F8/E6) = mg((P, - PNpP, = mg

{(P -

PJV

-

PPJI.

-

111 (9)

Thus the quantity (p - p,)/(l - pp,) gi ves both modulus and argument of the ratio (H,/II,)l’2.