Indian Journal of Radio & Space PhysicsVol. 19,June 1990, pp. 105-117

Angular approximations for waves in a cold magneto plasma

Henry G Booker*

Department of Electrical Engineering and Computer Sciences,University of California, San Diego, La Jolla, California 92093, USA

and

HariOm Vats

Physical Research Laboratory, Ahmedabad 380 009, India

Received 23 March 1989; revised received 19 January 1990

Budden [J Atmos & Te" Phys (GB),45 (1983) 213]opined that the traditional quasi-longitudinal (QL)and quasi-transverse (QT) approximations in magneto-ionic theory are wrong, A more general approachto these angular approximations, based on simplifyingthe radical in the expressions for the squares of therefractive indices of the ordinary and extraordinary waves, is described, showing that the new Buddenformulae involve substantially the usual approximations, together with an additional approximation,which, in general, should be avoided. The two versions are comparable at sufficiently high frequencies,but the Budden QL approximation is undesirably restrictive for the whistler wave and the Budden QTapproximation is virtually useless for the directed Alfven wave, whereas the traditional approach yieldsthe Gendrin theory of group propagation in the whistler band and the standard treatment of the directedAlfven wave presented in books on magnetohydrodynamics, The validity of long-established and widelyused applications of the QL and QT approximations is reaffirmed.

1 IntroductionThe theory of waves in a cold magnetoplasma,

known as the magneto-ionic theory in studies of ra­dio propagation in the ionosphere under the influ­ence of the Earth's magnetic field, involves a qua­dratic dispersion relation, with two solutions corre­sponding to two characteristic waves. These areknown as the ordinary wave (0 wave) and the ex­traordinary wave (X wave), although, in general,both waves are affected by the presence of the im­posed magnetic field. The theory employs the fol­lowing angular frequencies: W (wave frequency),wN (plasma frequency), ve and Vi (electronic andionic collisional frequencies), wMe and WMi (elec­tronic and ionic gyro-frequencies), WC1 and WC2

(lower and upper critical frequencies where the re­fractive index of the X wave vanishes), Woo I andwoo2 (lower and upper hybrid resonant frequencieswhere the refractive index of the X wave is infinite

for propagation perpendicular to the imposed mag­netic field), and Wax (the OX transition frequencywhere, in a collisionless magnetoplasma, the refrac­tive indices of the 0 and X waves are equal forpropagation perpendicular to the imposed magnetic

*since deceased

field), The last frequency is one at which, in an al­most collisionless magnetoplasma, a cross-connec­tion phenomenon in the dispersion curves for the 0and X waves occurs as the direction of phase propa­gation is turned away from the transverse direction(the direction perpendicular to the imposed magne­tic field), A similar cross-connection phenomenonalso occurs in an almost collisionless magnetoplas­ma at the plasma frequency as the direction of phasepropagation is turned away from the longitudinal di­rectio~ (the direction parallel or antiparal1e1 to theimposed magnetic field). Here use of a single vetocharacterize the collisions of electrons with heavyparticles is an approximation that applies only atfrequencies well above the ion gyro-frequency.

Propagation of the 0 and X waves in a homogene­ous magneto plasma has been studied extensively; ashort summary is given by Booker!. The propertiesof the magneto plasma may be described in terms ofits longitudinal, transverse and Hall susceptibilitycoefficients, KL, KT and KH respectively. For a colli­sionless magnetaplasma composed of electrons anda single ion species, the expressions for the suscepti­bility coefficients are

2

WN

K - -- (1)L- 2 .••W

105

INDIAN J RADIO & SPACE PHYS, JUNE 1990

where

,2 Character of angular approximationsThe refractive index n of a characteristic wave is a

symmetric function of 0p, and Budden employs an

approximation to n( O~) in the vicinity of Op = 0 thatensures that iPn/aOp at 0r=O is calculated withcomplete exactitude at all wave frequencies. How­ever, by definition, an approximation does not haveto evaluate anything with complete exactitude in allcircumstances. An approximation only has to makecalculations with reasonable accuracy in certain cir­cumstances, and this applies' as much to a2 nldOi asto any other quantity.

It would certainly not be surprising if an approxi­mation to n( 0,,) in the vicinity of Or= 0 evaluatedthe low-order derivatives of n at Or= 0 with reason­able accuracy. But there is no rigorous necessity forthis to happen. Consider, for example, a geometricalsurface that may be described as a pimple on apumpkin. If one is expanding about the angular po­sition of the centre of the pimple, the low-order de-

(iv) The quasi-transverse (QT) approximation in

which \he sin4 Op termynder the radical dominatesthe COS" Op term ..

QL and QT approximations are applicable notonly to the general dispersion relation in Eq. (4) butalso to the radio and hydromagnetic approxim­ations thereto l.2.

It is with the angular approximations (QL andQT) that this paper is concerned; one of the authors(HGB) first worked with these approximationswhile studying under J A Ratcliffe (Refs 3 and 4).The QL approximation is used for estimating colli­sional absorption in HF radio communications4-6,for studying whistler propagation 7-9, for calculatingFaraday rotation of the direction of polarization insatellite radio communications6•1O, and for describ­ing the ionospheric transmitted wave in ELF com­municationsll. The QT approximation is not usedmuch at radio frequencies but, below the ionic gyro­frequency, it provides the standard treatment of AIf­ven waves that appears in books on magnetohy­drodynamics.

Nevertheless, a paper has recently appeared sug­gesting that the traditional QL and QT approxim­ations are wrongl2• They are in fact reliable approx­imations if properly used. Nevertheless, it is truethat no comprehensive study of their ranges of val­idity seems to have been published. In this paper amore general approach to angular approximationsis employed that shows how the traditional approx­imations and the alternative angular approximationsrecently presented by Budden fit into an overall pic­ture.

... (3)

... (5)

... (2)

... (6)

... (4)

In addition, there are unrationalized versions ofEqs (4) and (6) in which the radical is shifted to thedenominator.

Various approximations exist to the general dis­persion relation in Eq. (4). In particular, these arethe following:

(i) The radio approximation, applicable for allvalues of Or at frequencies larger than the lower hy­brid resonant frequency.

(ii) The hydromagnetic approximation applicable

for all values of Or at angular frequencies smallerthan Min ((ON' (OMe)' The freq'Jency bands of validityfor the radio and hydromagnetic approximationsoverlap except for a low-density plasma.

(iii) The quasi-longitudinal (QL) approximationin which the cos2 Op term under the radical in Eq. (4)dominates the sin4 Op tcrm.

For a collisionless magnetoplasma, the radical inEq. (4) is real and positive; the upper sign refers tothe 0 wave and the lower sign to the X wave. Theangle Op is to be distinguished from Og, which is theangle made with the imposed magnetic field by thedirection of group propagation.

Equation (4) can also be written in the form

n2=(1+1<L)

In terms of 1(L, 1(T and 1(H, the refractive index n of acharacteristic wave whose direction of phase propa­gation make!';an angle Op with the imposed magneticfield is given by

106

I i ~ I I I '1111I Illi I. I II

BOOKER & VATS: ANGULAR APPROXIMATIONS FOR WAVES IN A COLD MAGNETOPLASMA

For a collisionless magnetoplasma the value' of ris obtained by substituting into Eq. (5) the values of1(L, 1(T and 1(H appearing in Eqs (1), (2) and (3). Theresult may be written conveniently with the aid ofthe OX transition angular frequency wax. We obtain

which may be approximated as2 2

. WMe(Wax - W )

r=j (2 2) ... (10)W WN-W

Figure 1 shows how I rl varies with wave frequen-

3 Transition angle between the QL and QTapproximationsThe transition from QL to QT behaviour occurs

where the two terms under the radical in Eq. (4) areequal. This happens if

Op = OLT or 7C - OLT ... (7)

where the transition angle OLT is given by

.,. (9)

... (8)sin2 (}LT _ J.

2 cas (}LT I rl

rivatives can give an excellent description of theshape of the pimple but provide essentially no infor­mation about the shape of the pumpkin. Quite goodapproximations to the surface of the pumpkin canexist that completely disregard the very existence ofthe pimple. Other approximations may exist that de­scribe both the pimple and the pumpkin with rea­sonable accuracy.

The pimple-and-pumpkin phenomenon is in­volved in QL and QT approximations to differentextents at different wave frequencies. We shall findthat the most dramatic example occurs at frequen­cies lower than the ionic gyro-frequency. For the QTapproximation to the directed Alfven wave, thefirst-order Budden approximation evaluates n 2 toan accuracy of 1 per cent over a range of Op less than1°, tending to zero as the wave frequency tends tozero. By contrast, the first-order traditional approx­imation evaluates n 2 to an accuracy of 1 per centover a range of Op around 55° at W = 10-1 WMi' andaround 85° at W = 10-2 WMi' tending to 90° as Wtenns to zero.

I" is clear that what is involved in QL and QT ap­prcximations requires more careful study than hasbeen given to them by Booker3, by Buddenl2 or byanyone.

Wax

o

-I10

10

tITI

(a)

w2 = 10-"2 W2N Me

WMi='0-4 WMe

10

~ 0

1021 •• ,\ /, , , , ~ ,8L--_.L..-_.L..-_L-.........lJL-.L-.J'----'-2 -I I -I 2 ~ -I I -I 2IOWMi IOWMi WMi IOwMllWMiWMe)2 lOw Me WMe lOw Me IOWMe IOWMi IOWMi wMi IOWMi (u.;.IWMer~ IOWMe WMe IOWMe IOwMe

ANGULAR WAVE FREQUENCY

Fig. 1-Dependence of I rl (lhe parameter that controls QUQT behaviour) on angular wave frequency for two collisionless magneto­plasmas

107

INDIAN J RADIO & SPACE PHYS, JUNE 1990

cy for two collisionless magneto plasmas in whichthe ratio of the electronic mass to the ionic mass is10- 4, and in which

... (11)

The two ionization densities illustrated in Fig. 1differ by a power of ten, the right half of the diagramreferring to the larger ionization density. We seethat, over the electromagnetic spectrum, the numer­ical value of I rJ varies over many powers of ten, sothat far more than the numerical value of 0p is in­volved in deciding whether it is the cos2 0p term orthe sin4 0p term under the radical in Eq. (4) thatdominates. Use in Eq. (8) of the numerical values ofI rl shown in Fig. 1 leads to Fig. 2, which shows thevariation of the transition angle 0LT separating QLfrom QT behaviour with angular wave frequency.We see that there are bands of frequency wherepropagation is QL for a wide range of values of 0p,and other bands where propagation is QT for a widerange of values of Op.

Because of the factor W6x - W 2 in the numeratoron the right-hand side of Eq. (9), propagation at theOX transition frequency in a collisionless magneto­plasma is QL for all values of 0p except for 0p = 90°.At w = wox there is a discontinuity in the behaviouras 0p -+ 90° and for Op = 90°. This discontinuity has

been discussed by Booker!. If collisions are incor­porated to a small extent, the curves in Fig. 2 do notquite touch the 0p = 90° edge of the diagram; nearW = Wox a rapid transition takes place in the disper­sion curves as 0p -+ 90°, and this involves cross-con­nection between the dispersion curves for the 0 andX waves at w = Wox.

Again, because of the factor %- W 2 in the den­ominator on the right-hand side of Eq. (9), propaga­tion in a collisionless magnetoplasma at the plasmafrequency is QT for all values of 0p except for0p = 0° and 180°. This discontinuity has also beendiscussed by Bookerl.l3.1f collisions are incorporat­ed to a small extent, the curves in Fig. 2 do not quitetouch the 0p = 0° edge of the figure; near W = wN arapid transition takes place in the dispersion curvesas 00 -+ 0°, and this involves the phenomenon ofcross-connection at W = wN.

4 Region ofvalidity for the first-order angularapproximations

The dispersion relation in Eq. (4) is not undulycomplicated, and in many circumstances the em­ployment of angular approximations is not worth­while. The main feature of the dispersion relationthat is sometimes worth simplifying is the radical,and it is tq,wards this that angular approximations

· ,/

I

60"~ I \f -160"

01 / \Itl(l) OLOL/OL \IOLI /'" ,\J ~ ep0. <D

LT

i--eLT

30"I /I

(a)

( b)

W2 = 10-1/2 W2

W2 _101/2 W2N Me

N - Me

WMi=1O-4 WMe

WMi =10-4 WMe

001 I

,

,,-2 -I

1 -I 2 -2 -II -I2

IOWMi IOWMiWMiIOWMi (wMiwMel~ IOWMe WMe IOWMe IOWMe IOWMi IOWMi WMiIOWMi (wt.1iWMe)~ IOWMe WMeIOWMeIOWMe

ANGULAR WAVE FREQUENCY

Fig. 2-Dependence of transition angle OLT between quasi-longitudinal (QL) and quasi-transverse (QT) behaviour on angular wavefrequency for two collisionless magnetoplasmas

108

I I ~I I" I I' 'JIll 1 II~I~ I II

BOOKER & VATS: ANGULAR APPROXIMATIONS FOR WAVES IN A COLD MAGNETOPLASMA

... (12)

are directed. The radical may be expanded by thebinomial theorem, and the series may then be trun­cated to provide an approximation. For QL behav­iour, expansion is in powers of

Tsin" 8p

2 cos 8p

QL approximation in Eq. (17) for W < wox and forW > WN becomes

2 _ ( )(1 + 1(T)- t 1(HTS1ll28p ± j1(Hlcos 8pln - 1+ 1(L • 2(1 + 1(d - (1(L- 1(T)sm 8p

'" (19)

... (17)

2 _ ( )(1 + 1(T)- + 1(HTsin28p =+= j1(H!COS8pln - 1+ 1(L • 2(1 + 1(L)- (1(L- 1(T)sm 8p

'" (20)

where both in Eqs (19) and (20), the upper signs ref­er to the 0 wave and the lower signs to the X wave.

The conditions for applicability of the Q Land QTapproximations may be illustrated in an (w, 8p) dia­gram by shading regions where the calculation of n 2

is accurate to better than about 1 per cent. We shalluse vertical shading for the QL approximation andhorizontal shading for the QT approximation. Fig. 3shows, on this basis, the regions of validity of thefirst-order angular approximations for the same twocollisionless magnetoplasmas used in Figs 1 and 2.The curves for the transition angle 8LT between QLand QT behaviour shown in Fig. 2 run between tliehorizontally and vertically shaded regions in Fig. 3.We see that there are substantial ranges of wave fre­quency in which either the first-order QL approxi­mation or the first-order QT approximation is use­ful over substantial ranges of angle .

but for wox < W < WN it becomes

5 Importance of avoiding angular approximationsthat upset an infinity of a refractive indexIt will be noticed that, in the preceding section, we

have made angular approximations in the numera­tor of Eq. (4) but not in the denominator. The suc­cess of angular approximations depends to an im­portant extent on this feature.

The reciprocal of the denominator in Eq. (4) or(6) could, if desired, be expanded by the binomialtheorem in ascending powers of expression (12) or(13), but it wo'uld be simpler to pe.rform the expan­

sion in ascending powers of sin2 8p or cos2 8r If thelatter is done, it would be necessary for sin 8p orcas2 8p to be less than the appropriate radius of con­vergence, and to be small compared with it if a cou­ple of terms are to constitute a good approxima­tion. The magnitude of this radius of curvature canbe calculated from the vanishing of the denominatorin Eg. (4) or (6). This corresponds to an infinite va­lue for the refractive index of one of the characteris­

tic waves, and therefore to an edge of a pass bandfor this wave. Before such an edge is approached for

... (13)

... (15)

... (16)

... ( IX)

and for QT behaviour, expansion is in powers of

2 cos 8p

. 2 8Tsm p

n2 = (1 + 1(L)1(1+ 1(T)- 1(HTsin2 8pl(1 + 1(T)+ (1(L- 1(T)cas 2 8p

... (14)

Second-order approximations in which thesquares of the quantities (12) and (13) are retainedare often not needed. Let us consider first-order ap­proximations in which the quantities (12) and (13)are retained but their squares are neglected. We thenobtain a QT approximation in the form

On the other hand, for the corresponding QL ap­proximation, we obtain

This gives for the 0 wave

2 (1 + 1(L)(1+1(T)n = 2

(1 + 1(T)+ (1(L- 1(T)cos 8p

and for the X wave

Care is necessary with the interpretation of thealternative signs in Eq. (17). If collisions are neglect­ed, the radical in Eg. (4) has its positive value. Con­sequently, Eq. (10) shows that if the cos 8p term inEg. (17) is interpreted as Icos 8pl, then its coefficientmust be interpreted as

whereas the T preceding the curly bracket in Eq.(17) must be interpreted as written in Eg. (10) with­out the use of absolute-value signs. Therefore, the

109

INDIAN J RADIO & SPACE PHYS, JUNE 1990

WlDt Wax Wile Wel WN Wal2 Wc2

---~ W~QT,

60

QL

..•..•."- ,

\\\\\\\\\I\\

\

IIIIII1III'

(a)

W2'~'l'0-1/2 W2N Me

-4 W

,"M"'IIII M,QL

90°

600

01<I>

"U...

Q.<D

ANGULAR WAVE FREQUENCY

Fig. 3-Regions in the (w, Or)plane where the first-order quasi-longitudinal approximation (vertical shading) and the first-order qua­si-transverse approximation (horizontal shading) evaluate the square of the refractive index for a characteristic wave to an accuracy

of about I per cent or better in two collisionless magnetoplasmas.

one of the characteristic waves, an approximationinvolving binomial expansion of the reciprocal ofthe denominator in Eq. (4) or (6) breaks down. Weshall see that this extra restriction on the range ofvalidity of the approximation is sufficiently seriousand that angular approximations to the reciprocal ofthe denominator in Eq. (4) or (6) need to be avoided.

The locations in Fig. 3 where the denominator inEq. (4) or (6) vanishes are indicated by brokencurves. The broken curve in the frequency bandW.::;; WMi runs from W = WMi at the bottom of the dia­gram to W = 0 at the top of the diagram, almostalong the line W = WMi and then almost along theaxis Or = 90°. It defines the edge of a pass band forthe 0 wave that extends from the broken curve

down to Or = 0° in angle and down to W = 0 in fre­quency. The nearness of the curve to the axisOr = 90° in Fig. 3 means that an approximation in­volving expansion of the reciprocal of the denomin­ator in Eq. (6) in ascending powers of cos2 Or wouldbecome invalid almost immediately upon departurefrom the axis Or = 90° (cos 0p = 0). It is essential toavoid this if, for the 0 wave, a useful range of valid­ity is to exist for the QT approximation at wave fre­quencies less than the ionic gyro-frequency.

The broken curve in Fig. 3 for the frequency band

Woo t'::;; W.::;; Min (wN, wMe) runs from w=Min (wN,

wMe) at the bottom of the diagram of W = wOOlat thetop of the diagram. It defines the edge of the passband for the whistler wave that extends from the

broken curve down to Or = 0° in angle and down toW = 0 in frequency. For the higher of the two ioniza­tion densities shown in Fig. 3, it will be noticed thatin the band wox < W < Min (wN, wMe) the first-or­der QL approximation derived in the preceding sec­tion only loses 1 per cent accuracy after stop-bandconditions for the whistler wave have been reached,whereas an approximation that involved expansionof the reciprocal of the denominator in Eq. (4) in as­cending powers of sin2 0p would become invalidwell before stop-band conditions are encountered.

The broken curve in Fig. 3 for the frequency bandMax (wN, wMe).::;; W < Woo 2 runs from W = Max (wN,

wMe) at the bottom of the diagram to the higher va­lue W = Woo 2 at the top of the diagram. It defines theedge of a pass band for the X wave that extends

from the broken curve up to 0p = 90° in angle anddown to W = we I in frequency. For the higher of thetwo ionization densities shown in Fig. 3, it will benoticed that in the band Max (wN, wMe)< W< wco2

the first-order QT approximation derived in thepreceding section only loses I per cent accuracy af-

t •

110

I I II I, 1-1' ~II, I I. I ",);, ,I.Id I. III" 1111111 • "

__ ~~_~o~~, __,

BOOKER & VATS: ANGULAR APPROXIMATIONS FOR WAVES IN A COLD MAGNETOPLASMA

... (22)

for Q1 behaviour. Incorporating these restrictions,the areas of the (w, Op) plane in which an accuracy ofabout 1per cent or better is obtained for both char­acteristic waves using first-order approximationsare shown in Fig. 4.

Comparison of Fig. 3 with Fig. 4 demonstratesthat the additional restriction (21) has little influenceupon the region of validity of the QL approximationin the frequency band W > Max (wN, wMe). In theband WMi < W < Min (wN, wMe), however, the in­equality (21) significantly restricts the region of val­idity of the QL approximation for the whistler wave.It does not restrict the region of validity of the QLapproximation for the other characteristic wave inthe band WMi < W < Min (wN, wMe) but, being nor­mally evanescent in the whistler phenomenon, thiswave is of limited practical interest. In the bandW < WMi' comparison of Figs 3 and 4 shows that theadditional restriction (21) does little damage to theQL approximation.

Comparison of Figs 3 and 4 also shows that, forthe QT approximation at frequencies in the vicinity

... (21)

for QL behaviour, and

ter stop-band conditions for the X wave have beenreached, whereas an approximation that involvedexpansion of the reciprocal of the denominator inEq. (6) in ascending powers of cos2 Op would be­come invalid well before stop-band conditions areencountered.

Let us now suppose that, instead of the QL andQT approximations derived in the preceding sec­tion, we arrive at alternative angular approximationsby expanding the right-hand side of Eq. (4) or (6) inascending powers of sin2 Op for QL behaviour, andin ascending powers of cos2 Op for 01' behaviour.We are then approximating not only the numeratorin Eq. (4) but, for a characteristic wave whose passband is bounded by one of the broken curves in Fig.3, also the reciprocal of the denominators in Eqs (4)and (6). In this way we obtain the formulae that havebeen presented by Buddenl2• From the denomina­tors in Eqs (4) and (6) we see that, for the relevantcharacteristic wave, this procedure involves the ad­ditional restriction

sin2 Op 411 + KL IKL - KT

QL

..•..•,,\

\\\\\\\\\IIIIIIIIIII

Wml (''''x Wile Wel WN WcD2Wc2 '"' ~-u __ ~ ~ J

QT

(b)

W~ =101/2W~e

WMi =10-4 WMe

GO°

Q

" ",,\\\\\\\

\

,,\,\

I\,I\I,IIII,IIIIIII

(a)

w2 = 10-1/2 W2N Me

WMi=10-4 WMe

0'1CII

"U"'

a.<D

90"

ANGULAR WAVE FREQUENCY

Fig. 4-Regions in the (w, (}p) plane where the first-order approximations obtained by expanding the right-hand side of Eq. (4) inpowers of sin2 (}p (quasi-longitudinal behaviour, vertical shading) and cos2 (}p (quasi-transverse behaviour, horizontal shading) evalu­ate the square of the refractive index for both characteristic waves to an accuracy of about 1 per cent or better in two collisionless

. magnetoplasmas.

111

INDIAN J RADIO & SPACE PHYS, JUNE 1990

tures of Op from 90°, the range of validity in wavefrequency is inconveniently narrow. It is usually ad­visable to avoid angular approximations near zerosof refractive index.

For the reasons described in the previous section,it continues to be important in general to make noangular approximation in the denominators of Eqs(24) and (25). The coefficient of the sin2 Op termdropped from the numerators of Eqs (24) and (25)behaves quite differently as a function of wave fre-

6 Regions of validity for angular approximationsof practical valueWhile the first-order QL and QT approximations

have the regions of validity illustrated in Fig. 3, it isoften convenient in practice to accept somewhatmore restricted ranges of validity as illustrated inFig. 5. Here the region of "validity for the QT ap­proximation is fully maintained for W ~ WMi, but therest of the region of validity for the approximation isdeleted. The two regions of validity for the QL ap­proximation are retained, but with less extensiveboundaries.

Figure 3 shows that, for much of the frequencyrange from about the ionic gyro-frequency upwards,the angular range of validity of the first-order QTapproximation is too restricted to be useful. An ex­ception occurs near the plasma frequency but here,for the larger departures of Op from 90°, validity ofthe QT approximation is too restricted in wave fre­quency to be convenient. For the X wave, the ap­proximation given in Eq. (16) is useful in the fre­quency range Max (wN, wMe) < W < Woo 2' but thisfrequency range is not of much practical interest forthe X wave. It is only in the band W ~ WMi that theQT approximation has genuine practical value.

While the first-order QL approximation given inEqs (19) and (20) may be used in the regions of val­idity illustrated in Fig. 3, it is often convenient toemploy the zero-order approximation' having themore restricted regions of validity illustrated in Fig.5. The zero-order QL approximation involves neg­lect of not merely the square of expression (12) butalso this expression itself. The sinz Op terms are thendropped from the numerators of Eqs (19) and (20),giving for W < wox and W > wN

.--r

.,,(24)

... (25)2 (1 + I<L)((1+ I<T) =+= jl<Hlcos Opl}

n = . z(1+I<L)-(I<L-I<T)sm Op

z _ (1 + I<d ((1 + I< T) ± j I< HIcos 0pi}

n - (1+I<L)-(I<L-I<T)sinzOp

and for wox < W < WN

". (23)

Validity of the angular approximations derived inthe preceding section then usually extends not onlyup to the edge of the pass band at n = co , but alsoeven into the stop band beyond. This is only true,however, if the restrictions (21) and (22) areavoided. In particular, it is not true for approxim­ations based on expanding Eqs (4) and (6) in as­cending powers of sin2 Op and cos2 Op'

In view of the success of the QL and QT approx­imations in functioning up to the edge of a pass bandassociated with an infinity of refractive index, it isappropriate to enquire about the edge of a passband associated with a zero of refractive index.Here the QL and QT approximations are consider­ably less successful. The conditions for n = 0 are in­dependent of the value of Op' In Fig. 3 the zeros ofrefractive index for the X wave occur on the ordi­nates W = WCl, wcz. Here, the angular ranges of val­idity for both the QL and QT approximations, al­though not negligible, are not large. The zero of re­fractive index for the 0 wave occurs on the ordinateW = wN• Here the angular range of validity of the QTapproximation is excellent but, for the larger depar-

of the plasma frequency, the inequality (22) restrictsthe region of validity of the approximation for the Xwave to some extent. There is no corresponding res­triction for the 0 wave but, because this wave has azero of refractive index at the plasma frequency, useof the QT approximation for this wave near this fre­quency is inconvenient. For the region of validity ofthe approximation below the ionic gyro-frequency,Fig. 4 exhibits a drastic reduction in comparisonwith Fig. 3. This reduction applies to the 0 wave,which is the directed Aliven wave. The reason that,for this wave, the effect of the restriction (22) is sodrastic is that high conductivity along the imposedmagnetic field normally makes I<Llarge; in the hy­dromagnetic approximation I<Lis taken as infinite.On the other hand, there is no corresponding res­triction for the X wave when W:::;; WMi. This is thewave that becomes the omnidirectiQnal Aliven waveas W -+ 0 and does not normally call for the use ofangular approximations.

The success of the angular approximations de­rived in the preceding section and illustrated in Fig.3 depends to a great extent on their ability to func­tion satisfactorily close to infinities of refractive in­dex. The greater the ionization density, the betterthey do this. This is illustrated in Fig. 3 by the factthat the broken curves are inside the shaded regionsin the right half of the diagram to a greater extentthan they are in the left half. It is quite common inpractice to have

112

I I H II • ,litI 1 III· II 11

BOOKER & VATS: ANGULAR APPROXIMATIONS FOR WAVES IN A COLD MAGNETOPLASMA

QL

"...,\

\\

\Top OF

\ WHISTLER

\BAND\\I,\,\I,I,III

QL

(b)

W~ =101/2W~e

WMi =10-4 WMe

QLQL

(0)

W2 =10-1/2 W2N Me

WMi=10-4 WMe

'",\,\\-"

\ TOP OF\ WHISTLER

\ BANO\\,I\,,,II,IIII,IIIIII•

0'" • , 1111111111111111, l, 11111111111 00

-2 -I 1 -I 2 -2 -I 1 -I 2IOWMi 10WMi WMi IQwMi (WMiwMel2 IOwMe WMe 10wMe 10 WMe IOWMi IOWMi WMi IOWMi (WMiWMel! I0WMe WMe lOwMe IOWMe

90°

a.CD

ANGULAR WAVE FREQUENCY

Fig. 5-Regions in the (w, Op) plane where it is convenient in practice to employ the zero-order quasi-longitudinal approximation(vertical shading) and the first-order quasi-transverse approximation (horizontal shading) for calculating the square of the refractive

index for a characteristic wave to an accuracy of about 1 per cent or better.

quency from the coefficient of the sin2 Op term re­tained in the denominators. In the vertically shadedregion of Fig. 5 located above the angular frequencyMax (wN, wMe), one has 1(L -:- 1(T, so that the sin2 Op

term in the denominator is then unimportant. But inthe vertically sbaded region of Fig. 5 located in thewhistler band, the sin2 Op term in the denominatorof Eq. (25) plays the significant role of defining thelocation of the infinity of refractive index and theedge of the whistler band, while the sin2 Op term thathas been dropped from the numerator is unimport­ant because I rl is then small as illustrated in Fig. 1.

7 Accuracy of iJ2 n/80/ using angular approxi­mations

The calculation of 82 n/8 (j2 assumes importancebecause this is required for the purpose of evaluat­ing the distant field of a source embedded in ahomogeneous magnetoplasma.

The first-order QL approximation [Eqs (19) and(20)] evaluates 82 n/80/ at Op = 0° without error.However, the zero-order QL approximation (Eqs24 and 25) only evaluates 82 n/80p2 at Op = 0° to afractional accuracy of approximately'>l rl. From Fig.1 we can see that, at the bottom right-hand corner ofeach part in Fig. 5, 82 n/80p2 is evaluated by thezero~order QL approximation to an accuracy of

about 1 per cent. At higher wave frequencies the ac­curacy is better but, as the frequency descends toleft-hand edge of the vertically shaded region locat­ed above the angular frequency Max (wN, wMe), theerror in the evaluation of 82 n/80p2 at Op = 0° in­creases to about 10 per cent. In the vicinity of theOX transition angular frequency Wax in Fig. 5, thezero-order QL approximation evaluates 82 n/80/at 0 = 0° to an accuracy of better than 1 per cent.However, in Fig. 5, at the two extreme edges of thevertically shaded region located in the whistlerband, the zero-order QL approximation only evalu­ates 82 n/80p2 at 0 = 0° to an accuracy of about 10per cent. Nevertheless, everywhere in the verticallyshaded regions of Fig. 5, n 2 is evaluated to an accu­racy of about 1 per cent Of better.

The second-order QT approximation evaluates82 n/80/ at Or = 90° without error. However, thefirst-order QT approximation (Eqs 15 and 16) onlyevaluates 82 n/80/ at Op = 90° tb an accuracy of ap­proximately 21 rl- 2. From Fig. 1 we can. see that, atthe right-hand edge of the horizontally shaded re­gion in Fig. 5, 82 n/80/ is evaluated at Op = 90° toan accuracy of about 2 per cent. The accuracy im­proves rapidly as the frequency is reduced.

For a source embedded in a homogeneous mag­netoplasma, the distant field radiated in the strictly

113

INDIAN J RADIO & SPACE PHYS, JON E I990

8 QT approximation when OJ ~ OJMi

The Alfven refractive index IlA is defined by

Let us consider a magnetoplasma for which the

density is high enough to make IlA ~ 1. WhenOJ ~ OJMi,it is true a fortiori that OJ ~ ( OJMeOJMY /2 ,

OJ ~ OJMeand OJ ~ OJN• Moreover, the last two in­

equalities are extremely well satisfied. Consequent­ly, Eqs (1), (2) and (3) become approximately

longitudinal and strictly transverse directions maybe evaluated using the QL and QT approximations

to the accuracy just described for a21l/a()/ at()p= (t and 90° respectively. However, for calculat­ing the complete radiation polar diagram of a source

embedded in a magneto plasma, a21l/a()/ must beevaluated with reasonable accuracy for the whole

range of values of ()p that correspond to pass-bandbehaviour. For this purpose angular approximationsin any form are unsatisfactory. For calculating theradiation polar diagram of a source embedded in ahomogeneous magnetoplasma, unapproximatedformulae should be used.

'., -

9 QL approximation when OJ ~ Max (OJN, OJMe)

In the vertically shaded region of Fig. 5 locatedabove the angular frequency Max (OJN, OJMe)we mayemploy the zero-order QL approximation given inEq. (24). In this formula we may use the radio ap­proximations to 1<L' 1<T and 1<H obtained by puttingOJMi= 0 in Eqs (I), (2) and (3). We obtain, when(OJM/ OJ J2 is negligible,

2 21<l = - WN/ W •. , (33)

As already mentioned in Sec. 2, QT behaviour inthe frequency band OJ ~ OJMidramatically illustratesthe effect of the additional restriction that is incor­

porated in the approximation of Budden 12 and thatis implied by the inequality (22). At wave frequen­cies of the order of the lower hybrid resonant fre­quency and below, the conductivity of a high-dens­ity magnetoplasma parallel to the imposed magneticfield is large. This makes 1<L large in the inequality(22), and reduces the right-hand side to a value ofthe order of 10 - 4 or less at wave frequencies lessthan the ionic gyro-frequency. Consequently, for thedirected Alfven wave, the Budden approximation

only describes a local peculiarity near ()p= 90°, andmisses the practically important result given in Eq.

(31). The peculiarity near ()p= 90° upon which Bud­den concentrates is completely disregarded in Eq.(31) because of its minimal practical interest. How­ever, Eq. (15) is quite capable of retaining the pecu­

liarity near ()p= 90°, if desired. It is merely a matterof using in Eq. (15) the exact expression for 1<L ap­pearing in Eq. (1) instead ot the approximate ex­pression appearing in Eq. (27).

'". (30)

... (27)

... (28)

... (29)

... (26)

1<L = 00o

1<] = IlA

• 2 /1<H = - JIlA OJ OJMi

while Eq. (5) becomes approximately

T =j OJM/ OJ

11; = I+

These are the expressions for 1<L' 1<T, 1<H and T to beused in Eqs (15) and (16) for the QT approximationwhen OJ ~ OJMi.We obtain for the 0 wave

Eqs (31) and (32) are the formulae obtained forthe refractive indices of the characteristic waves if

the wave frequency is allowed to tend to zero in theunapproximated formulae appearing in Eqs (1 )-(6).Eqs (31) and (32) show that the same formulae maybe used throughout the horizontally shaded regionin Fig. 5 to calculate 11 2 for the two characteristicwa~es to an accuracy of about 1 per cent or better.The QT approximation appearing in Eqs (31 ) and(32) constitutes the treatment given in books onmagneto hydrodynamics for the directed and om­nidirectional Alfven waves.

This is not quite the traditional QL approxima­tion for the band w ~ OJMe.The latter is obtained byworking from the unrationalized version of Eq. (4),and is written as

... (37)

... (34)

... (36)

... (35)

, w~/ (JJ 2n-= 1----------

1 ± (w!'tfe/w)lcos Opl

1<H = - j~wMe/OJ

and substitution into Eq.i(24) then gives

n2 = 1 = (~/W2)\1 + (wMe/w)lcos Opl}

However, use of the binomial theorem shows that,

when (OJM/ OJ)2 is negligible, Eqs (36) and (37) areconsistent. For further discussion, see the work ofHeadingl4.

Equations (36) and (37) illustrate the fact that, for

... (31)

... (32)

11 = IlAlsec ()pl

and for the X wave

114

I , I!

BOOKER & VATS: ANGULAR APPROXIMATIONS FOR WAVES IN A COLD MAGNETOPLASMA

'" (38)

portant when wN ~ wMe' so that Eqs (42) and (43)are consistent. For further discussion, see the workof Headingl4•

In the lower part of the frequency band betweenthe two gyro-frequencies we need to use the hy­dromagnetic approximation rather than the radioapproximationl•2. Eqs (1), (2) and (3) then become,for a high-density plasma,

." (48)

... (44)

... (45)

... (46)

... (47)

2

2 %n =w(wMelcos Bpl- w)

"T= "Hr-1"H = j WN2 I( WWMe)

and substitution into Eq. (24) gives

n2 = ~ 1 [=1= 1+ j r ]WWMeleas Bpi Icos Bpi

where r is given by Eq. (10). Because Irl is small inthe middle and lower parts of the frequency bandWMi~ W ~ WN (Fig. 1), Eq. (47) does not differmuch from what is obtained by extending Eq. (42)down to the bottom of the band and switching thealternative signs at the OX transition angular fre­quency wax.

The upshot for the whistler wave is that, in thepart of its pass band for which W ~ wMi,we have forthe zero-order QL approximation, when WN ~ WMe,

the zero-order QL approximation, behaviour of n2depends only on the component of the imposedmagnetic field parallel to the direction of phasepropagation. Equations (36) and (37) are valuablefor calculating Faraday rotation of the direction ofpolarization in satellite radio communications.When el~ctronic collisions are taken into account,the formulae are also valuable for calculating colli­sional absorption in HF radio communications. Forthe latter purpose we replace wi in Eq. (37) by

w,i{l - j( vel W)} -I and WMe by WMe {I - j (vel W)} -I.

The rate of attenuation per unit distance, a (in nep­ers), in the direction of phase propagation is thenevaluated, for sufficiently small values of Ve, as

1/2W

x {w(w ± wMelcos BpI)_ ~}1/2

The presence of collisions increases the degree ofvalidity of the QL approximation.

10 QL approximation when WMi ~ W < wMe ~ WN

In the upper part of the frequency band betweenthe two gyro-frequencies we may use the radio ap­proximation to Eqs (1),(2) and (3). We obtain, when(wi WMe)2is negligible,

"L=-~/w2 ... (39)

From the vanishing of the denominator in thisequation we see that a useful approximation to thelocation of the edge of the whistler band in an (w,

Bp)diagram when wN ~ wMe is given by

W = wMelcos Opl .. , (49)

11 Comparison of the zero-order QL approxi-mation with the unapproximated formulaewhen WOOl < W < WMe ~ WN

It is upon Eq. (48) that Gendrin8 based his treat­ment of group propagation of the whistler wave inthe frequency band above the lower hybrid resonantfrequency. The magnitude of the group velocity cal­culates to

U= C(WMe!wN)lsec0pl[(wlwMe)

X {Icos Opl- (wi wMe)}{(l+ 3 cos2 Op)

- 81cos Opl(wi WMe)+ 4( wi WMe)2}]1/2... (50)

and the angle 08 between the direction of grouppropagation and the imposed magnetic field is givenby

'" (42)

... (43)

'" (40)

... (41)

2

n2=1----%-W (w =1= wMelcos BpI)

The unity in this equation is numerically unim-

22 %n =-

W (w =1= wMelcos BpI)

The traditional QL approximation in this band isnormally derived from the unrationalized version ofEq. (4). It avoids the assumption that W ~ WMiand iswritten as

These values of "L' "T and "H are to be substitut­ed into Eq. (25) for the zero-order QL approxima­tion. The important practical case is that in whichinequality (23) is satisfied, and we then obtain ap­proximately

115

INDIAN J RADIO & SPACE PHYS, JUNE 1990

... (52)

_ sin Opllcos Opl-2(w/WMe)} ()tan Og - 2 ••• 51(1+cos Op)-2IcosOpl(w/WMe'

Let us use Eqs (50) and (51) to compare, for a col­lisionless magnetoplasma, calculations made usingthe zero-order QL approximation with calculationsmade using Eqs (1 )-(6) without any approximations.Let us use an ionization density corresponding tothe right halves of Figs 1-5, so that

WN = 101/4 WMe

For this ionization density, inequality (23) is notwell satisfied, but we shall nevertheless employ Eqs

(48 )-(51 ). Moreover, we shall use these formulae upto the top of the whistler pass band shown in theright half of Fig. 5. We are therefore using the zero­order QL approximation not only inside the verti­cally shaded region for the whistler band in the righthalf of Fig. 5 but also outside it as far as the brokencurve. In these circumstances one might expect thatagreement between the approximate and the'exacttreatment would be somewhat marginal.

In Fig. 6 the magnitude U of the group velocity isexhibited in a polar diagram as a function of theangle Ogof group propagation, with the angle Opofphase propagftion used as a parameter along the

0

-2-2-2 '-OAxlO c 0.8x10 c1.2xlOc 0O.05eO.lOeO.l5co.2Oe

IIII

~QIOe-2 o Q4xl0 c...J

II t=: W =O.99wMe I~~w=0.70WMeLLJ i:i: iO.05cQ i=LLJZ(.!)

°IS ~Bo )1~

)+ -f0<[ 80~

0LLJtJ> 1\l....- I~'"" ..fi--foo5c0 11-~ -2o OAxlO e~ I --T I~ -fO.lOe

a:::<[...J::>Q Q2c0 z

I II -10.leLLJ 11-W =O.l9wMew =0.02WMea:::

I:LJ11-O.le

>-~<..:l9

O~'t:iIs d---------r -f0

LLJ-.

:>-11-::>0a::: O.1e(.!)

I II IQle

III I

0.2e .,..... ..,

0O.lcO.2cQ3co.4e0- O.lcO.2eO.3e

GROUP VELOCITY PARALLEL TO IMPOSED MAGNETIC FIELD

Fig. 6-Group propagation of the whistler wave for a series of frequencies in the radio band for a collisionless magnetoplasma. Eachpanel is a polar diagram showing how the magnitude U of the group velocity varies with the angle 8g that the group velocity vectormakes with the imposed magnetic field Bo, which is directed horizontally. An arrow at the tip of a group velocity vector (U, 8g) indi­cates the corresponding angle 8p that the direction of phase propagation makes with Bo. The upper half of each panel is based on theapproximate dispersion relation in Eq. (48), while the bottom half is based on the unapproximated dispersion relation in Eq. (4).

Here w~ = 101/2 wJ:e.

116

I I ~, I I' '11111 II" I' I II

I

iI

BOOKER & VATS: ANGULAR APPROXIMATIONS FOR WAVES IN A COlD MAGNETOPLASMA

curve. In each panel the imposed magnetic field Bois directed horizontally. The three-dimensional po­lar diagram of group velocity is obtained from thetwo-dimensional polar diagram by rotation about anaxis through the origin S parallel to Bo and by reflec­tion in a plane through S perpendicular toBo. Theparametric values of Op are indicated by means ofarrows for which the counter-clockwise angle withthe direction of Bo is Op' The physical significance ofdiagrams such as those shown in Fig. 6 for radiationfrom a source S has been discussed by Gendrin8,Booker and Dyce2 and Booker! .

Each panel of Fig. 6 contains two polar diagrams,an upper one and a lower one, that are approximate­ly symmetrical. The upper half is calculated on thebasis of the approximate dispersion relation given inEq. (48). The lower half is calculated on the basis ofthe unapproximated dispersion relation given in Eq.(4). The lack of symmetry between the upper andlower halves of each panel indicates the error in­volved in the approximation.

We notice that, even for propagation in the strictlylongitudinal direction, the group velocity is a littledifferent in the two halves of each panel in Fig. 6.This does not arise from use of the zero-order QLapproximation given in Eq. (25). It arises from useof the inequality (23) for an ionization density. that infact satisfies Eq. (52). As the ionization density is in­creased above the value corresponding to Eq. (52),the pairs of polar diagrams of group velocity com­pared in Fig. 6 become increasingly symmetrical.

The highest three frequencies depicted in Fig. 6lie entirely outside the vertically shaded region forthe whistler band in the right half of Fig. 5. It is onlythe bottom right-hand panel in Fig. 6, drawn forW = 0.02 WMe, that involves points within theshaded region, and then only up to about Op = 86°.The stop band is entered when Op reaches the value88.8°. It is the values of Op from about 86° to 88.8°that account for the asymmetry between the approx­imate and the exact polar diagrams at the far right ofthe bottom right-hand panel in Fig. 6. It is remark­able how well the zero-order QL approximationperforms.

The bottom left-hand panel in Fig. 6 correspondsto locations for the whistler band in the right half ofFig. 5 that lie just outside the vertically shaded re­gion. The top right-hand panel corresponds to loc­ations appreciably outside, and the top left-handpanel to locations well outside. Nevertheless, in nocase does the zero-order QL approximation encou­rage misleading physical concepts; This is an illus­tration of the fact that, for most practical purposes,the zero-order QL approximation and the first-or­der QT approximation are usable over regions in

Fig. 5 larger than those indicated by the vertical andhorizontal shadings respectively.12 Conclusion

It is concluded that it is inappropriate to suggestthat the traditional approach to QL and QT approx­imations is wrong. Above the electronic gyro-fre­quency, the traditional approach and the new Bud­den approach have comparable degrees of validity.Any substantial difference between the two types ofapproximation in this band is an indication that theunapproximated formulae should be used. But,when one descends to the whistler band, the Bud­den QL approximation is more restrictive than thetraditional one. Gendrin8 would not have been able

to develop his theory of group propagation in thewhistler band if he had been required to work withthe Budden QL approximation. When one descendsfurther in frequency to the Alfven band, the BuddenQT approximation for the directed Alfven wave isnot just embarrassingly restrictive; it is disastrouslyrestrictive unless the density of the magnetoplasmais so low that the velocity of Alfven waves is com­parable with the velocity of light in free 'Space. TheBudden QT approximation for the directed Alfvenwave concentrates on a trivial feature usually disre­garded in practice, whereas the traditional approachgives the standard treatment presented in books onmagneto hydrodynamics.13 Acknowledgement

This work was supported by National ScienceFoundation, USA, through a grant ATM81-06147atUCSD.References

1 Booker H G, Phi/os Trans R SOl' London A (GB), 280(1975) 57.

2 BookerH G& DyceRB, RadioSci(USA),69D(1965)463.3 Booker H G, Proc R SOl' London Ser A (GB), 150 (1935)

267.

4 Ratcliffe J A, The magneto-ionic theory and its applicationsto the ionosphere (Cambridge University Press, Cambridge),1959.

5 Appleton E V, Proc R SOl' London Ser A (GB), 162 (1937)451.

6' Davis K, Ionospheric radio propagation (National Bureau ofStandards, U S Government Printing Office, Washington,DC),1965.

7 Storey L R 0, Phi/os Trans R SOl' London A (GB), 426(1953) 113.

8 Gendrin R, Planet &Space Sci( GB), 5 (1961) 274.9 Helliwell R A, Whistlers and related ionospheric phenomena

(Stanford University Press, Stanford, USA), 1965.10 Little C G & Lawrence R S, J Res Natl Bur Stand(now Radio

Sci( USA), 64D (1960) 335.11 Booker H G & Lefeuvre F, J Atmos & Terr Phys (GB), 39

(1977) 1277.12 Budden K G, J Atmos&Terr Phys(GB), 45 (1983) 213.13 Booker H G. Proc R SOl' London Ser A (GB), 147(1934)

352.

14 HeadingJ, JAtmos & TerrPhys (GB), 46 (1984) 1169.

117