angular approximations for waves in a cold magneto...
TRANSCRIPT
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