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Quasielectrostatic whistler-mode propagation
II032 0633:85 I3 on + 0 00 J: 1985 PcrgamonPress Ltd
QUASIELECTROSTATIC
WHISTLER-MODE
PROPAGATION
S.S. SAZH1.Y Private Scientist
: prospectKosmonavtov
18-2-44, Leningrad
196244. U.S.S.R.
and E. M. SAZHINA Physical Institute, State University,
(Receiwd
infinulfirm
Leningrad 3 Sepremher
198904, U.S.S.R. 1984)
Abstract -An approximate formula is derived for the refractive index ofa whistler-mode wave propagating in a hot anisotropic plasma with wave normal angle close to the resonance cone angle(0,). Approximations used during the derivation are generally satisfied for magnetospheric conditions. It is pointed out that the derived formula can be considered to be complementary 10 the corresponding formula for quasilongitudinal whistlermode propagation in a hot anisotropic plasma which was derived by Sazhin and Sazhina (1982). The limits of applicability of a cold plasma model when determining the height of generation of saucer emissions and V-shaped hiss are discussed.
1. INTROL)t;CTION
Since the appearance of Storey’s (1953) pioneering work, thousands ofpapers which in some way concern the problem of whistler-mode propagation and interactions in the magnctospheric plasma have been published (see the reviews of Hclliwcll, 1965; Gendrin. 1975 ; Sazhin, 1982a, and Anderson, 19X3). During the last few years whistler-mode waves in the solar wind and in other planets’ magnetospheres have also been studied (see Anderson, 1983 ; Coroniti et al., 1982; Inan et crl., 1983). Experimental stud& of whistler-mode waves have strongly stimulated the development of different aspects ofthcir theory. Amongst the problems which were most cxtensivcly studied we can mention different aspects of linear whistler-mode instabilities (Hascgawa, 1975 ; Cuperman, 19X1: Gcndrin, I98 I ; Salhin, lYX2b). nonlinear interactions ofelcctrons with monochromatic whistler-mode waves (Karpman. lY74; Karpmanrtal., 1Y74a. b; KarpmanandShklyar. lY77), quasilincar theory of whistler-mode waves (Bcspalov and Trahtengertz, 1980; Gendrin. 1981 : Sa7hin. 19X4a), whistler-mode propagation in the magnetosphere (Walker. 1976; Strangcways and Rycroft, I980 : Strangeways, 1% I. I982a. b ; Karpman and Kaufman. 1981. 1982, lYX3a.b; Hudden. 1983: Stott, 1983). At the same time. in most of these studies it was assumed that whistler-mode propagation could bc described by a cold plasma model [among the abovementioned papers a review paper of Cupcrman (1081) probably presents the only exception]. which is not always justified for magnetospheric conditions (Hess.
196X ; Akasofu and Chapman, 1972). Some attempts were made to consider the influence of the electrons’ finite tcmpcrature and anisotropy on whistler-mode propagation in the magnetospheric plasma by using numerical methods (Willis, 1975; Hashimoto and Kimura, 1973, I9X I ; Cuperman, 19XI ; Moreira, 1982 ; Kobclev and Sazhin, 1983a; Sazhin, 1984b) but the domain of applicability of these approaches remained limited. Hence, it seemed necessary for us to develop some general analytical models for whistler-mode propagation in a hot anisotropic plasma when keeping in mind magnetospheric conditions (Sazhin, 1981, 1983; Sazhin e/ al.. I981 ; Sazhin and Sazhina. 1982; Kobelcv and Sarhin, I983b, 19X4). The latter approach made it possible not only to justify the limits of applicability of the cold plasma approximation, but also todiscover somenew propcrtiesofthesc wavesand to present a new method of approach to the problem, useful for further research. At the same time, this analysis has been limited so far to waves with wave normal angles not close to the resonance cone angle, and this is a serious restriction of the theory. since for actually obscrvcd whistler-mode waves in the magnetospherc this condition is not always satisfied (see Maggs. 1976, 197X; Alckhin and Shklyar, 1980: Hashimoto and Kimura, 19X1 ; llayakawa et al., 19X4). llcncc, the consideration of the problem of the propagation of whistler-mode waves in a hot anisotropic plasma with wave normal angles close to rcsonancc cone angle Ox. using analytical methods, seems to be rclevanl and will be considered in this paper. Following the suppositions of Sazhin and
S. S. SAZHIN and E. M. SAZHINA
296
Sazhina (1982) we shall restrict our analysis to consideration of whistler-mode propagation in a dense plasma (electron gyrofrequency R is well below the electron plasma frequency II); the obtained formula will then appear to be complementary to that of Sazhin and Sazhina (1982). The principal formulae are derived in Section 2. In Section 3 the approximations made in our analysis are considered in detail. The analysis of the formulae obtained, when considering ma~etospheric applications, is presented in Section 4. The main new results of the paper are summarized in Section 5.
2. THEORY
The dispersion equation valid for the general case of electromagnetic wave propagation in a magnetized homogeneous plasma can be presented in the form (Akhiezer et al., 1975) :
proportional to w: or wf Explicit formulae for ajf) are presented by Sazhin et al. (1981), Sazhin (1981), Sazhin and Sazhina (1982). After inserting (5) into the corresponding formulae for A, B and C and separating the terms which either do notdependonplasmatemperature,orareproportional to w: or wf,we can present the expressions for A, B and C in the form A = A,+A, B = B,+B,
(6)
c = c,+c,
where A,, B, and Co are the corresponding coefficients determined within the cold plasma model (see Akhiezer et al., 1975) : W(d - Q2co2 0) .WZ(R2- w2)
A,=lf
(7)
B _ _2 + II*[S2Z-4w*+QZCos28] AN4fBN2+C
= 0
(1)
where N is the wave refractive index, the coefficients A, B and C are functions of the components of the dielectric tensor fij and wave normal angle 8; explicit formulae for these are given by Akhiezer et al. (1975) and Sazhin and Sazhina (1982). The expressions for the components ofEijin their turn depend on the choice of particle distribution function. Following Dory, Guest and Harris( 1965), weemploy in our further analysis the electron distribution function in the form : f=
(j!~3~2w~j+Zw,,)-1u~exp (-J+$)
(2)
[where w ,,(We) is the electron thermal velocity in the direction along (perpendicular) to the external magneticfield;j = 0,1,2,3,. . .] ;protonmotionwillnot be considered. The same model was also employed in our previous papers (see for example Sazhin et al., 198 1; Sazhin, 1981; Sazhin and Sazhina, 1982). Explicit expressions for aij using this model were derived by Sazhin et al. (198 1). in the same work these expressions were further simplified when inequalities :
0-
o.G(!Y-w2)
-
2I-F
w2(sP- 03’) (8)
+ c 0
= 1 + I12(302 -U) .~w*(Q2- ru2)
3114 w2(& - 02) +
IF
04(cz2- f2) . (9)
Coefficients A,, B, and C, which are proportional to wi or w: have in general a rather complicated form; for our further analysis we need the explicit form of only the first of them : A, = ~N2~z~~w2(~2-~‘)3(4r;zZ-~2)]-1 x (co54 B[( - 2wa + 2206@ - 67w4Q4 + 470*R6
-
12@) + A,(2ws - 220%~
+ 32w4Q4- 12~*@)] + co? 0 x [( - UP - L&22 + 22w4ff4 - 8wW) + A,@*
+ 16w6Q2 - 29w4Q4 + 12wa@)-J
+ A,( - 3w8 + 6w6Q2 - 3m4R4)}
5. = I@ - a/k k$wij2Q2
(,w ,/I >>1
(3)
cc 1
(4)
(10)
where : p = (i/;?)wfi I-P/&Y
(11)
were valid, and presented in the form : f-., ‘U = #?‘+e!!’ I, 1,
A, = (j + l)w:/wi ;
(3
where w is the wave frequency; kllcs, is the parallel (perpendicular) component of the wave vector ; n = 0; & 1; +2; ~19)are the elements of the cold plasma dielectric tensor, and ai: )are the components of a tensor
w
c is the velocity of the light. When considering (6), (1) can be presented in the form : (A~+A*)N4-t(Bo+B~)N2~(C~+C,)
=O.
(13)
291
Quasielectrostatic whi stler-mode propagation We restrict our further analysis to consideration of whistler-mode propagation at wave normal angles close to the resonance cone angle 6,. When the plasma is cold, then the value of N becomes infinitely large when @approaches 0,. We can expect that within the considered plasma model, N will also be large enough in this region of 0 to neglect in equation (13) the terms which are not proportional to N2. Moreover, when keeping in mind that for sufficiently low temperature we have lBr1 CC/B& we can also neglect in the same equation the term B,N2 when compared with B,N* and present it in the form : a,/3N4+ AoN
+B,
in which the term Ai is presented
= 0
B, = 2r14/[W2(c22 -cl?)]
II, = arc cos (w/n).
(19)
a, = a,+u,O where : 02( - 2w4 + 1 lw2i12 - 5cP) a0 = ___
Q2(*2
(14)
_3)2
02(2w4 -
+A,
17w%P + 9R4) .fk2(R2 - w2) (4fF - 02)
(21)
0.9
where ai is the coefficient which explicit form obviously follows from (10). The condition (B,J CCl&,1 which has been used when deriving (14) is violated if B, is equal to zero simultaneously with A,. The latter can take place only in a raretied plasma (when w = II and 0, -+ 0) (Kobelev and Sazhin, 1984) which will not be concerned in our paper (see Introduction). This condition will be guaranteed by the inequalities (3)-(4). Equation (14) for the isotropic plasma was derived and its general properties were investigated by Akhiezer et al. (1975). In our further analysis, when following the general ideas of this monograph, we will apply this equation to the problem of whistler-mode propagation near the resonance cone in a hot anisotropic plasma. Keeping in mind that equation (14) reduces to the electrostatic wave dispersion equation if we neglect the term B0 in it, it seems justified to consider equation (14) as a quasiele~trostati~ whistler-mode dispersion equation and the waves which are determined by this equation as quasielectrostatic whistler-mode waves. This definition was also suggested by Hashimoto and Kimura (1981), although in a slightly different sense. We shall further restrict our analysis by considering wave propagation in a dense plasma (II/C! >P 1). This supposition can be violated in some regions of the equatorial magnetosphere outside the plasmasphere (Curtis, 1978) as well as in the aurora1 topside ionosphere (Maeda, 1975). In most regions of the magnetosphere, however, it seems to be justified (Hess, 1968); it enables us to present the formula for 0, in a most simple form : (16)
B as :
0=0,+-o
A, = 28’I12J(w,/i)Z)
as :
A, = a,/%N’
When considering
where 0’ << 1 (all the angles are expressed in radians), we can also simplify the formulae for ,4,, Be and a, [see equations(7), (8),(fO)and(l5)] and present themin the form :
(17)
( - 4w6 f 27w4fi2 - 27~~52~ + 8CZ6) X--(&_&)” +A
e
2oJGGP CP
(40~-39w4~2+410%4-12sP) x-(ccl2- w2)2(4!Y - fU2)
.
(22)
After substituting (17)-(22) into (14) we can obtain the soiution of the Iatter equation in the most simple form : N2 =
c”;&/w;
(23)
where
&==H’&J~
(25)
“+” in the last formula refers to 6” < 0; “-” refers to 0’ > 0; the choice of the sign when 0’ = 0 depends on the side from which one approaches this point [this sign convention makes it possible to reduce our solution (23) to the cold plasma solution N2 = -&,/A, when \j + 0] ; when deriving (23) we neglect the contribution of the terms proportional to fY/j’and 0” as well as the smaller terms. Let us first consider a, > 0 which implies &, < 0. In such a case (23) presents a real solution (N2 > 0) only when te cc 0. From (25) it follows that &+ is real only when
lo’/ > m;
being negative
when
0’ < -,/&$ and positive when 0’ 2 a. Thus we can conclude that the solution of (23) for a, > 0 exists only when u’ < -m or @d 0,-m. Let us now consider a, -=z0, which implies f. > 0. In
298
S. S. SAZHIN and E. M. SAZHINA
such a case (23) presents a real solution only when ts > 0 which is possible only for 8 < 0 or 8 < OR. When a, = 0 formula (23) is no longer valid (see next section). From this preliminary analysis it follows that formula (23) presents no real solution when 0’ > 0 or 0 > OR and we can further conserve only the sign “+” in it, that is, to consider it for 0’ < O(a, < 0) or 8’ < -a (a0 > 0). Our results are different from that which were obtained by Hashimoto and Kimura (1981) who claimed that temperature effects enable whistler mode propagation even for 0 > OR(see p. 11, 149 of their paper). It seems that this discrepancy comes from the fact that Hashimoto and Kimura used not quasielectrostatic but really electrostatic approximation when considering whistler-mode propagation, corresponding to the equation A = 0 [see equation(l)], and they did not use the restrictions (3)-(4) in their analysis. Experimental results ofHayakawa et al. (1984) seem to confirm our conclusion about whistler-mode propagation only at 6 < 0,. The plots a, vs w/n are presented in Fig. 1 for A, = 1 (solid) and A, = 2 (dashed). As follows from this figure a, < 0 for w < 0.69Q when A, = 1 and for o < 0.65Q when A, = 2. For these values of w we can expect that quasielectrostatic whistler-mode waves exist up to B = 0,. When o > 0.69R (A, = 1) or w > 0.65a (A, = 2) quasielectrostatic whistler-mode wavesexist only when
When presenting our previous analysis we made several suppositions which were not at first obvious. Before any further search for applications of the obtained formula (23), it seems necessary to consider these suppositions in some detail.
3. SUPPOSITIONS
When deriving the formula (23) we assumed the validity of (3)-(4) as well as the inequalities N2 D 1, 8’ << 1 and a@ CCa,. Inequality (3) for the case of whistler propagation@ < Qisvalidfor all nifit is valid for n = 0 and n = 1. Hence after substituting (23) into (3)-(4) we can present them in the form of a system of three inequalities :
&&o”/(n”(n-
coy) <<1
&&B2/Q2 <<1 &JLQu2(Q2 - 034,,/2n4
(26) (27)
<< 1
where A,, = w:/wi = A,/(j+ 1). When deriving (26)-(28) we took into account
(28) that
k;i N N2c04/c2Q2 k: u N2c02(C12 - 02)/S12c2.
To estimate the domain of the validity of(26)-(28) we can take the largest value of l&l equal to m. Substituting (24) into (26)-(28) we can join these three inequalities into a single one :
4
<
(29)
where Icl =
Jlaol(tko)2JiF7/(2JZw3)
K* =
JlaolJZG7/(2Jzo)
Kj = Jlaol.
a0
0
-1 FIG. 1. PLOTS
a,
VS
o/n
FOR
A, = 1
(DASHED).
(SOLID)
AND
A, = 2
CP(j + l)/(JZA,wJZZ7).
Plots K1,2,3 are presented in Fig. 2 for A, = 1 (solid) and A, = 2 (dashed). j is supposed to be equal to zero ; generalization of our analysis for j # 0 is straightforward. From Fig. 2 it follows that (29) is never satisfied for w z 0.69R (A, = 1) or w = 0.65R (A, = 2), that is for those frequencies when a, = 0. Outside the vicinity of these frequencies the range of w for which (29) is valid strongly depends on A,. When A, = 1 and w < 0.69R (29) is satisfied for larger fl than in the case when A, = 2 in the same frequency range and vice versa for w > 0.69Q. Keeping in mind that for the real magnetospheric conditions /3 is in most cases (with the exceptions of magnetosheath and plasma sheet conditions) less than 0.02, we can expect (26)-(28) to be valid for A, = 1 and w 5 0.6a.
Quasiel~trostatic Ki
K3
KI
K3
299
whistier-mode propagation 1
“3
FIG. 3.
PLOTS
tlo,ke
VS W/a
FOR z‘f, =
1 (SOLID)
AND
A,
= 2
(DASHED).
FIG. 2. PLOT‘sicf,Z,S VS co/ii
FOX
A, = f (SOLID)AND
A,
=
2
(DASHED).
o i= 0.6R. In order to assess them for a particular case one can use the curves plotted in Figs. 2 and 3. Let us consider inequality N2 >> 1. When considering (23)-(25) we can obtain the expression for the maximum value of N2 (which is attained at B = BRwhen a, 0) in the form : JlGlm* N&x = 1501-
n2/Q2.
(30)
Taking into account that l&,1> 1; la,( P /I (or laol >>/S (when w is not small) and II* >>f2’ we can see from the latter formula that N&, >> I. From (25) we can see that this inequality is satisfied not only for N2 = Ni, but also for other values of N2 if B is not too far from 8,: 10’15 lo--t?,l << 1. The last inequality was also supposed to be valid when deriving (23). In particular, it was explicitly used when we neglected the terms proportional to B’@in the derivation of (23). Strictly speaking, when making the last operation we required not If71c 1 but it71CClao/ael. In order to control the latter inequality we plotted in Fig. 3 the curves [a,/~,1 vs w/Q for A, = 1 (solid) and A, = 2 (dashed), As follows from this figure the restriction on 0 in order that our model could be valid, is less stringent for A, = 1than for A, = 2. In the vicinity of the points where ae = 0 our model also is violated because of the restriction on 0’. Thus it follows from the analysis of this section that the suppositions used when deriving the formula for whistler refractive index in the form (23) are not contradictory and can be simultaneousIy satisfied for realistic magnetospheric conditions, at least for
4. DISCUSSION
As was already pointed out in Section 3, the value of NZ attains its largest value in the vicinity of 8 = 19,for aoi:OorB=B,a for a. > 0 determined by (30), contrary to the case ofwhistler-mode propagation in a cold plasma where N2 + 03 when B -+ 8,. Let us now present an expression for NiBX in a more convenient form :
As follows from the latter form&a Ni, --+co when wii -+ 0. K&X also becomes infinitely large when a, -+ 0 but in this case our model is no longer valid. It can be easily proved that in the whole range of 8 considered, N2 is a monotonically increasing function of 8. We now consider the problem of wave energy propagation when based on the derived formula (23). However before proceeding in this direction, we recall some fundamental points of the theory of wave propagation (Budden, 1961; Six, 1962 ; Helliwell, 196.5).We presented in Fig. 4 a schematic plot of a part of the curve N(B) in polar coordinates (solid segment) for 0 close to 19,.Axis N,, as well as the dashed segment on the same figure coincides with the direction of the external magnetic field B,,. A perpendicular to the curve
S.S.SAZHIN and E. M.
300
VOh /
N, FIG.~. SCHEMATIC PRESENTATION 0~ em DIRECTIONS V, AND v, CORRE.WONDlNG TO THE SEGMENT OF THE CURVE M(e) (SOLID).
N(O) at the point corresponding to 8 = 8, indicates the direction ofwave group velocity which forms an angle $ with the magnetic field. An angle between V, and V,, we labelled, following Stix (1962), 8,; it is determined by the equation (Stix, 1962) : Jtan @,I= IN-r*dAJ/dBl.
(32)
After substituting (23) into (32) we have : (tan @,I= (2Jm)
- ‘.
(33)
Let us first consider the case when a0 > 0. From (33) it follows that \tan 8,1-+ cc when I#i -+ @ which implies that [es] -+ 7r/2for these B’,similarly to whistlermode propagation in a cold plasma. The value of II/($,,) in this case is determined by the formula : $c, = n/2-8,tJ2a,P.
(34)
The value of #, determined by (34) is larger than the corresponding value of $, for whistler-mode propagation in a cold plasma at 6 close to ORwhen $,, YE$, = R/2-&. For a, < 0 it follows from (33) that ltan @,I4 l/ ,,/?$a when 8’ -+ 0. Hence the value of $, is determined by the equation : $,, = larc~an(UJGiiiF)--~~l.
(35)
The value of $, determined by (35) is obviously less than $,. The case a, < 0 corresponding to o/Q < 0.65 when A, = 2 (or 0.69 when A, = 1) seems to be more typical for whistler-mode waves observed in the magnetosphere (see Sazhin, 1982a). it will further be of our primary attention. The determination of the values of $ in the vicinity of B = e, is especially important for several magnetospheric applications. We will consider briefly one of
SAZHINA
them. The value of $,, was used when determining the height of generation of saucer emissions and V-shaped hiss registered by satellites in the topside ionosphere (Mosier and Gurnett, 1969). When making their estimates these authors used a cold plasmamodel and it isnot obviousapriorithat thismodelisvalidfor thereal magnetospheric conditions. In order to check .its validity let us calculate $,, when using (35). Supposing that II/C2 = 3 ;mewi/ = 1 eV we have: /? = 1.8 lo- ‘. Taking further o = 0.4Q A, = 1, and, correspondingly a, = -0.44, we have arctan(l/~) = 89.5”. Hence it seems justified for us to use the methods suggested by Mosier and Gurnett (1969) when determining the height of the region of the considered emissions generation within the cold plasma model. At the same time the influence of the electrons’ finite temperature and anisotropy on the value of $,, seems to be much more important in the equatorial magnetosphere. Let us consider for example plasmaspheric conditions: II/Q = 100; mew;/2= 1 eV; /I = 0.02. Taking again o = 0.44 A, = 1 and correspondingly laoI = 0.44, we have: arctan (l/s) = 75.1”. Hence $, determined by (35) is equal to X.7”,which is less than the corresponding value of $C = 23.6”. Our analysis of Section 3 made it possible to justify the validity of our formula(23) for wave normal angles 0 close to 6a (or 0,-a). In order to control its validity for B away from ORwe presented in Fig. 5 the curves N(B) calculated from the formula (23) (solid), from the equation (13) (dotted), and from equation (14) where no restrictions on 0 were imposed (dashed), when supposing that w = 0.4!& II = lOR, mew!/2 = 10 eV (,Q= 0.002). Parts of these curves are presented in Fig. 6, when using another scale of N,, axis. As follows from these curves the plots calculated from (23) and (13) practically coincide when B 2 38,/4. At the same time for B < 30,/4 the curve calculated from (13) practically coincides with the corresponding curve calculated from the formula derived by Sazhin and Sazhina (1982) [see their formula (25)] even within the accuracy of Fig. 6. Thus our formula (23) can be considered as complementary to that which has been derived by Sazhin and Sazhina (1982). Since the value of N calculated from the latter formula tends to decrease when 6 approaches O,, we can expect that for a certain 0 = 0, the curves calculated from (23) and from the formula of Sazhin and Sazhina (1982) intersect. Thus when modelling whistler-mode propagation in the real plasma we can set N(0) =
N,(e) N,(e)
0G 02
0, e,
(36)
where N r(@is determined by formula(25) of Sazhin and
301
Quasielectrostatic whistler-mode propagation
FIG. 5. PLOTS N(o) CALCULATED FROM THE FORMULA (23) (SOLID),FORMULA (13) (DOTTED), FOKMULA (14) (DASHED);DOTTED CURVE c01~clDEs WITHIN THE ACCURACY 0~ A PLOT, WITH THE CORRESPONDING cuwE CALCULATEDPROMFORMULA (25) OR SAZHINAND SAZHINA (1982).
Sazhina (1982), N,(H) is determined by formula (23) of our paper. The complementary character of the curves N,(O) also occurs for A, = 2 and other values of o/n and /I. it is important to notice that the curve calculated from (14) which at first sight better approximates the value of i?(n), appears to be slightly worse than that which was
calculated from (23). [Error which comes from presenting equation (13) in the form (14) seems to be partly compensated by the error coming from the condition 0’ c< 1.1 Hence it seems sufficient to use (23) when approximating N(B) even if f3 is not very close to 8,. 5.
(a) We have derived a formula for N(B) for whistlermode propagation in a hot anisotropic plasma valid when 8 is close to 0,. This can be considered as complementary to the corresponding formula for quasilongitudinal whistler-mode propagation which was derived by Sazhin and Sazhina (1982). (b) The derived formula describes whistler-mode propagation similar to that for a cold plasma for 0 < OR but, contrary to the case of whistler-mode propagation in a cold plasma, it predicts finite N when B approaches B,. (c) From the derived formula, it follows that the electrons’ finite temperature and anisotropy influence the direction of V, when 6 is close to 0, ; this influence is especially important for the equatorial regions of the magnetosphere where the value of /I is not so small (- 10m2) compared for instance with its value in the aurora1 topside ionosphere (- IO-‘).
t
4
---%_____Nc'
I 10
I
REFERENCES
50
NJ FIG.~.
CONCLUSIONS
THEPARTSOFTHE PLOTS PRESENTEDON ANOT~SRSCALEOF~ ,, AXIS.
FIG.~ WITH
Akasofu, S.-I. and Chapman, N. S. (1972) Solar-Terrestrial Physics. Clarendon Press, Oxford. Akhiezer,
A. I., Akhiezer,
1. A., Polovin,
R. V., Sitenko,
A. C.
302
S. S. SAZHIN and
and Stepanov, K. N. (1975) Plasma Electrodynamics. Pergamon Press, Oxford. Alekhin, Yu. K. and Shklyar, D. R. (1980) Some problems of VLF waves propagation in the magnetosphere. Geomugn. Aeron. 20,501 (in Russian). Anderson, R. R. (1983) Plasma waves in planetary magnetospheres. Rev. Geophys. 21,414. Bespalov, P. A. and Trahtengertz, V. Yu. (1980) Cyclotron instability in the Earth’s radiation belts, in Problems of Plasma Theory (Edited by Mihalovskii, A. B.), pp. 88-163. Atomizdat, Moscow (in Russian). Budden, K. G. (1961) Radio Waues in the ionosphere. Cambridge University Press, Cambridge. Budden, K. G. (1983) Approximations in magnetoionic theory. .I. atmos. terr. Phys. 45,213. Coroniti, F. V., Kennel, C. F., Scarf, F. L. and Smith, E. J. (1982) Whistler mode turbulence in the disturbed solar wind. J. geophys. Rex A87,6029. Cuperman, S. (1981) Electromagnetic kinetic instabilities in multicomponent space plasma: Theoretical predictions and computer simulation experiments. Rev. Geaphys. 19, 307. Curtis, S. A. (1978) A theory for chorus generation by energetic electrons during substorms. J. yeophys. Res. 83, 3841. Dory, R. A., Guest, G. E. and Harris, E. G. (1965) Unstable electrostatic plasma waves propagating perpendicular to a magnetic field. Phys. Rev. Left. 14, 131. Gendrin, R. (1975) Waves and wave-particle interactions in the magnetosphere: a review. Space Sei. Rev. 18, 145. Gendrin, R. (1981) General relationships between wave amplification and particle diffusion in a magnetoplasma. Rev. Geophys. 19, 171. Hasegawa, A. (1975) Plasma Instabilities and Nonlinear E’cts. Springer, Berlin. Hashimoto, K. and Kimura, I. (1973) Oblique whistler mode propagation in a hot plasma. Rep. ionosp~. Space Res. Japan 27,73. Hashimoto, K. and Kimura, I. (198 1) A generation mechanism of narrow band hiss emissions above one half the electron cyclotron frequency in the outer magnetosphere. J. geophys. Res. 86, 11.148. Hayakawa, M., Yamanaka, Y., Parrot, M. and Lefeuvre, F. (1984) The wave normals of magnetospheric chorus emissions observed on board GEOS-2. J. geophys. Res. 89, 2811. Helliwell, R. A. (1965) Whistlers and Related Ionospheric Phenomena. Stanford University Press. Stanford, California. Hess, W. N. (1968) The Radiation Belt and Magnetosphere. Blaisdell, Waltham, Massachusetts. Inan, U. S., Helliwell, R. A. and Kurth, W. S. (1983)Terrestrial versus Jovian VLFchorus : a comparative study. J. geo~~~~~s. Res. 88,6171. Karpman, II. I. (1974) Nonlinear effects in the ELF waves propagation along the magnetic field in the magnetosphere. Space Sci. Rev. 16, 361. Karpman, V. I. and Kaufman, R. N. (1981) Tunnel transformation of whistler-mode waves in inbomogeneous plasma. J. exp. theor. Phys. SO, 1845 (in Russian). Karpman, V. I. and Kaufman, R. N. (1982) Whistler wave propagation in density ducts. J. Plasma Phys. 27,225. Karpman, V. I. and Kaufman, R. N. (1983a) On the peculiarities of whistler-mode propagation in the magnetosphericducts in the vicinityoftheequatorial region. I. Ducts with enhanced density. Geomagn. Aeron. 23, 451 (in Russian).
E. M. SAZHINA Karpman, V. I. and Kaufman, R. N. (1983b) On the peculiarities of whistler-mode propagation in the magnetospheric ducts close to the equatorialregion. II. Ducts with reduced densitv. Geumaan. Aeron 23,791 (in Russian). Karpman, V. I., Istomin, ?a. N. and Shklyar, D. R. (1974a) Nonlinear theory of a quasimonochromatic whistler mode nacket in inhomogeneous ulasma. Plasma Phys. 16,685. Karpman, V. I., Is&mitt, Ya: N. and Shklyar, D. R. (1974b) Nonlinear frequency shift and self-modulation of the quasimonochromatic whistler in the inhomogeneous plasma (magnetosphere). Planet. Space Sci. 22,859. Karpman,V.I.andShklyar,D. R.(l977)Particleprecipitation caused by a single whistler-mode wave injected into the magnetosphere. Planet. Space Sci. 25, 395. Kobelev, V. V. and Sazhin, S. S. (1983a) An estimate of magnetospheric electron temperature from the form of whistlers’ spectrograms. Letters to J. Tech. Phys. 9,862 (in Russian). Kobelev, V. V. and Sazhin, S. S. (1983b) On whistler-mode ’ trapping in the magnetospheric ducts. Geomagrl. Aeron. 23, 804 (in Russian). Kobelev, V. V. and Sazhin, S. S. (1984) Quasilongitudinal whistler-mode propagation in a rarefied plasma. Geophys. J. 6, 69 (in Russian). Maeda, K. (1975) A calculation of aurora1 hiss with improved models for geoplasma and magnetic field. Planet. Space Sci. 23, 843. Maggs, J. E. (1976) Coherent generation of VLF hiss. J. g&phys. Res. 81, 1707. Mamas. J.E.fl978)Electrostaticnoiseaeneratedbv theauroral e&ron beam.‘J. geophys. Res. 83,-3173. _ Moreira, A. (1982) On the influence of a plasma hot component on whistler propagation beyond the plasmapause. J. atmos. terr. Phys. 44,869. Mosier, S. R. and Gurnett, D. A. (1969) VLF measurements of the Poynting flux along the geomagnetic field with the Injun-5 satellite. 1. geophys. Res. 74, 5675. Sazhin,S.S.(1981)Somestudiesofwhistlermodepropagation in the magnetospheric plasma. J. atmos. terr. Phys. 43, 139, Errata, p. 373. Sazhin, S. S. (1982a) Natural Radio Emissions in the Earth’s Magnetosphere. Nauka, Moscow (in Russian). Sazhin, S. S. (1982b) A physical model for oblique whistlermode instabilities. Ann. Ge~p~ys. 38, 111. Sazhin, S. S. (1983) Whistler-mode propagation at frequencies _ near the electron gyrofrequency. J. Plasma Plzys. 29,217. Sazhin. S. S. (1984a) A model for hiss-twe mid-latitude VLF emissions. ‘Planer. Space Sci. 32, 126%. Sazhin, S. S. (1984b) On whistler-mode trapping in the magnetospheric ducts. J. Plasma Phys. 31,487. Sazhin, S. S., Kobeleva,O. A., Sazhina, E. M. and Varshavskii, S. P. (1981) On whistler-mode propagation at a small angle to the magnetic field in a hot anisotropic. plasma. Radio Phys. 24,922 (in Russian). Sazhin. S. S. and Sazhina. E. M. (1982) Obliaue whistler-mode propagationinahotanisotropicplasma,~. PlasmaPhys.27, 199. Stix, T. (1962) The ‘I’lzeory o~f’lasma Waves. McGraw-Hill New York. Storey, L. R. 0. (1953) An investigation of whistling atmospherics. Phi/. Trans. Roy. SW A246, 113. Stott, G. E. (1983) Refractive index surfaces. J. atmos. terr. Phys. 45, 219. Strangeways, H. J. (1981) Trapping of whistler-mode waves in ducts with tapered ends. J. atmos. terr. Phys. 43, 1071. Strangeways, H. J. (1982a) Investigation by ray-tracing of the
Quasielectrostatic
whistler-mode
effects of a summer-winter asymmetry on whistler ducting. J. atmos. terr. Phys. 44,889. Strangeways,H. J.(1982b)Theeffect ofmulti-ductstructureon whistler-mode wave propagation. J. atmos. terr. Phys. 44, 901. Strangeways, H. J. and Rycroft, M. J. (1980) Trapping of
propagation
303
whistler-waves through the side ofducts. J. atmos. terr. Phys. 42, 983. Walker, A. D. M. (1976) The theory of whistler propagation. Rev. Geophys. 14,629. Willis, J. W. (1975) Temperature effects on whistler-modes near gyroresonance. J. geophys. Rex 80, 1354.
QUASIELECTROSTATIC
WHISTLER-MODE
PROPAGATION
S.S. SAZH1.Y Private Scientist
: prospectKosmonavtov
18-2-44, Leningrad
196244. U.S.S.R.
and E. M. SAZHINA Physical Institute, State University,
(Receiwd
infinulfirm
Leningrad 3 Sepremher
198904, U.S.S.R. 1984)
Abstract -An approximate formula is derived for the refractive index ofa whistler-mode wave propagating in a hot anisotropic plasma with wave normal angle close to the resonance cone angle(0,). Approximations used during the derivation are generally satisfied for magnetospheric conditions. It is pointed out that the derived formula can be considered to be complementary 10 the corresponding formula for quasilongitudinal whistlermode propagation in a hot anisotropic plasma which was derived by Sazhin and Sazhina (1982). The limits of applicability of a cold plasma model when determining the height of generation of saucer emissions and V-shaped hiss are discussed.
1. INTROL)t;CTION
Since the appearance of Storey’s (1953) pioneering work, thousands ofpapers which in some way concern the problem of whistler-mode propagation and interactions in the magnctospheric plasma have been published (see the reviews of Hclliwcll, 1965; Gendrin. 1975 ; Sazhin, 1982a, and Anderson, 19X3). During the last few years whistler-mode waves in the solar wind and in other planets’ magnetospheres have also been studied (see Anderson, 1983 ; Coroniti et al., 1982; Inan et crl., 1983). Experimental stud& of whistler-mode waves have strongly stimulated the development of different aspects ofthcir theory. Amongst the problems which were most cxtensivcly studied we can mention different aspects of linear whistler-mode instabilities (Hascgawa, 1975 ; Cuperman, 19X1: Gcndrin, I98 I ; Salhin, lYX2b). nonlinear interactions ofelcctrons with monochromatic whistler-mode waves (Karpman. lY74; Karpmanrtal., 1Y74a. b; KarpmanandShklyar. lY77), quasilincar theory of whistler-mode waves (Bcspalov and Trahtengertz, 1980; Gendrin. 1981 : Sa7hin. 19X4a), whistler-mode propagation in the magnetosphere (Walker. 1976; Strangcways and Rycroft, I980 : Strangeways, 1% I. I982a. b ; Karpman and Kaufman. 1981. 1982, lYX3a.b; Hudden. 1983: Stott, 1983). At the same time. in most of these studies it was assumed that whistler-mode propagation could bc described by a cold plasma model [among the abovementioned papers a review paper of Cupcrman (1081) probably presents the only exception]. which is not always justified for magnetospheric conditions (Hess.
196X ; Akasofu and Chapman, 1972). Some attempts were made to consider the influence of the electrons’ finite tcmpcrature and anisotropy on whistler-mode propagation in the magnetospheric plasma by using numerical methods (Willis, 1975; Hashimoto and Kimura, 1973, I9X I ; Cuperman, 19XI ; Moreira, 1982 ; Kobclev and Sazhin, 1983a; Sazhin, 1984b) but the domain of applicability of these approaches remained limited. Hence, it seemed necessary for us to develop some general analytical models for whistler-mode propagation in a hot anisotropic plasma when keeping in mind magnetospheric conditions (Sazhin, 1981, 1983; Sazhin e/ al.. I981 ; Sazhin and Sazhina. 1982; Kobelcv and Sarhin, I983b, 19X4). The latter approach made it possible not only to justify the limits of applicability of the cold plasma approximation, but also todiscover somenew propcrtiesofthesc wavesand to present a new method of approach to the problem, useful for further research. At the same time, this analysis has been limited so far to waves with wave normal angles not close to the resonance cone angle, and this is a serious restriction of the theory. since for actually obscrvcd whistler-mode waves in the magnetospherc this condition is not always satisfied (see Maggs. 1976, 197X; Alckhin and Shklyar, 1980: Hashimoto and Kimura, 19X1 ; llayakawa et al., 19X4). llcncc, the consideration of the problem of the propagation of whistler-mode waves in a hot anisotropic plasma with wave normal angles close to rcsonancc cone angle Ox. using analytical methods, seems to be rclevanl and will be considered in this paper. Following the suppositions of Sazhin and
S. S. SAZHIN and E. M. SAZHINA
296
Sazhina (1982) we shall restrict our analysis to consideration of whistler-mode propagation in a dense plasma (electron gyrofrequency R is well below the electron plasma frequency II); the obtained formula will then appear to be complementary to that of Sazhin and Sazhina (1982). The principal formulae are derived in Section 2. In Section 3 the approximations made in our analysis are considered in detail. The analysis of the formulae obtained, when considering ma~etospheric applications, is presented in Section 4. The main new results of the paper are summarized in Section 5.
2. THEORY
The dispersion equation valid for the general case of electromagnetic wave propagation in a magnetized homogeneous plasma can be presented in the form (Akhiezer et al., 1975) :
proportional to w: or wf Explicit formulae for ajf) are presented by Sazhin et al. (1981), Sazhin (1981), Sazhin and Sazhina (1982). After inserting (5) into the corresponding formulae for A, B and C and separating the terms which either do notdependonplasmatemperature,orareproportional to w: or wf,we can present the expressions for A, B and C in the form A = A,+A, B = B,+B,
(6)
c = c,+c,
where A,, B, and Co are the corresponding coefficients determined within the cold plasma model (see Akhiezer et al., 1975) : W(d - Q2co2 0) .WZ(R2- w2)
A,=lf
(7)
B _ _2 + II*[S2Z-4w*+QZCos28] AN4fBN2+C
= 0
(1)
where N is the wave refractive index, the coefficients A, B and C are functions of the components of the dielectric tensor fij and wave normal angle 8; explicit formulae for these are given by Akhiezer et al. (1975) and Sazhin and Sazhina (1982). The expressions for the components ofEijin their turn depend on the choice of particle distribution function. Following Dory, Guest and Harris( 1965), weemploy in our further analysis the electron distribution function in the form : f=
(j!~3~2w~j+Zw,,)-1u~exp (-J+$)
(2)
[where w ,,(We) is the electron thermal velocity in the direction along (perpendicular) to the external magneticfield;j = 0,1,2,3,. . .] ;protonmotionwillnot be considered. The same model was also employed in our previous papers (see for example Sazhin et al., 198 1; Sazhin, 1981; Sazhin and Sazhina, 1982). Explicit expressions for aij using this model were derived by Sazhin et al. (198 1). in the same work these expressions were further simplified when inequalities :
0-
o.G(!Y-w2)
-
2I-F
w2(sP- 03’) (8)
+ c 0
= 1 + I12(302 -U) .~w*(Q2- ru2)
3114 w2(& - 02) +
IF
04(cz2- f2) . (9)
Coefficients A,, B, and C, which are proportional to wi or w: have in general a rather complicated form; for our further analysis we need the explicit form of only the first of them : A, = ~N2~z~~w2(~2-~‘)3(4r;zZ-~2)]-1 x (co54 B[( - 2wa + 2206@ - 67w4Q4 + 470*R6
-
12@) + A,(2ws - 220%~
+ 32w4Q4- 12~*@)] + co? 0 x [( - UP - L&22 + 22w4ff4 - 8wW) + A,@*
+ 16w6Q2 - 29w4Q4 + 12wa@)-J
+ A,( - 3w8 + 6w6Q2 - 3m4R4)}
5. = I@ - a/k k$wij2Q2
(,w ,/I >>1
(3)
cc 1
(4)
(10)
where : p = (i/;?)wfi I-P/&Y
(11)
were valid, and presented in the form : f-., ‘U = #?‘+e!!’ I, 1,
A, = (j + l)w:/wi ;
(3
where w is the wave frequency; kllcs, is the parallel (perpendicular) component of the wave vector ; n = 0; & 1; +2; ~19)are the elements of the cold plasma dielectric tensor, and ai: )are the components of a tensor
w
c is the velocity of the light. When considering (6), (1) can be presented in the form : (A~+A*)N4-t(Bo+B~)N2~(C~+C,)
=O.
(13)
291
Quasielectrostatic whi stler-mode propagation We restrict our further analysis to consideration of whistler-mode propagation at wave normal angles close to the resonance cone angle 6,. When the plasma is cold, then the value of N becomes infinitely large when @approaches 0,. We can expect that within the considered plasma model, N will also be large enough in this region of 0 to neglect in equation (13) the terms which are not proportional to N2. Moreover, when keeping in mind that for sufficiently low temperature we have lBr1 CC/B& we can also neglect in the same equation the term B,N2 when compared with B,N* and present it in the form : a,/3N4+ AoN
+B,
in which the term Ai is presented
= 0
B, = 2r14/[W2(c22 -cl?)]
II, = arc cos (w/n).
(19)
a, = a,+u,O where : 02( - 2w4 + 1 lw2i12 - 5cP) a0 = ___
Q2(*2
(14)
_3)2
02(2w4 -
+A,
17w%P + 9R4) .fk2(R2 - w2) (4fF - 02)
(21)
0.9
where ai is the coefficient which explicit form obviously follows from (10). The condition (B,J CCl&,1 which has been used when deriving (14) is violated if B, is equal to zero simultaneously with A,. The latter can take place only in a raretied plasma (when w = II and 0, -+ 0) (Kobelev and Sazhin, 1984) which will not be concerned in our paper (see Introduction). This condition will be guaranteed by the inequalities (3)-(4). Equation (14) for the isotropic plasma was derived and its general properties were investigated by Akhiezer et al. (1975). In our further analysis, when following the general ideas of this monograph, we will apply this equation to the problem of whistler-mode propagation near the resonance cone in a hot anisotropic plasma. Keeping in mind that equation (14) reduces to the electrostatic wave dispersion equation if we neglect the term B0 in it, it seems justified to consider equation (14) as a quasiele~trostati~ whistler-mode dispersion equation and the waves which are determined by this equation as quasielectrostatic whistler-mode waves. This definition was also suggested by Hashimoto and Kimura (1981), although in a slightly different sense. We shall further restrict our analysis by considering wave propagation in a dense plasma (II/C! >P 1). This supposition can be violated in some regions of the equatorial magnetosphere outside the plasmasphere (Curtis, 1978) as well as in the aurora1 topside ionosphere (Maeda, 1975). In most regions of the magnetosphere, however, it seems to be justified (Hess, 1968); it enables us to present the formula for 0, in a most simple form : (16)
B as :
0=0,+-o
A, = 28’I12J(w,/i)Z)
as :
A, = a,/%N’
When considering
where 0’ << 1 (all the angles are expressed in radians), we can also simplify the formulae for ,4,, Be and a, [see equations(7), (8),(fO)and(l5)] and present themin the form :
(17)
( - 4w6 f 27w4fi2 - 27~~52~ + 8CZ6) X--(&_&)” +A
e
2oJGGP CP
(40~-39w4~2+410%4-12sP) x-(ccl2- w2)2(4!Y - fU2)
.
(22)
After substituting (17)-(22) into (14) we can obtain the soiution of the Iatter equation in the most simple form : N2 =
c”;&/w;
(23)
where
&==H’&J~
(25)
“+” in the last formula refers to 6” < 0; “-” refers to 0’ > 0; the choice of the sign when 0’ = 0 depends on the side from which one approaches this point [this sign convention makes it possible to reduce our solution (23) to the cold plasma solution N2 = -&,/A, when \j + 0] ; when deriving (23) we neglect the contribution of the terms proportional to fY/j’and 0” as well as the smaller terms. Let us first consider a, > 0 which implies &, < 0. In such a case (23) presents a real solution (N2 > 0) only when te cc 0. From (25) it follows that &+ is real only when
lo’/ > m;
being negative
when
0’ < -,/&$ and positive when 0’ 2 a. Thus we can conclude that the solution of (23) for a, > 0 exists only when u’ < -m or @d 0,-m. Let us now consider a, -=z0, which implies f. > 0. In
298
S. S. SAZHIN and E. M. SAZHINA
such a case (23) presents a real solution only when ts > 0 which is possible only for 8 < 0 or 8 < OR. When a, = 0 formula (23) is no longer valid (see next section). From this preliminary analysis it follows that formula (23) presents no real solution when 0’ > 0 or 0 > OR and we can further conserve only the sign “+” in it, that is, to consider it for 0’ < O(a, < 0) or 8’ < -a (a0 > 0). Our results are different from that which were obtained by Hashimoto and Kimura (1981) who claimed that temperature effects enable whistler mode propagation even for 0 > OR(see p. 11, 149 of their paper). It seems that this discrepancy comes from the fact that Hashimoto and Kimura used not quasielectrostatic but really electrostatic approximation when considering whistler-mode propagation, corresponding to the equation A = 0 [see equation(l)], and they did not use the restrictions (3)-(4) in their analysis. Experimental results ofHayakawa et al. (1984) seem to confirm our conclusion about whistler-mode propagation only at 6 < 0,. The plots a, vs w/n are presented in Fig. 1 for A, = 1 (solid) and A, = 2 (dashed). As follows from this figure a, < 0 for w < 0.69Q when A, = 1 and for o < 0.65Q when A, = 2. For these values of w we can expect that quasielectrostatic whistler-mode waves exist up to B = 0,. When o > 0.69R (A, = 1) or w > 0.65a (A, = 2) quasielectrostatic whistler-mode wavesexist only when
When presenting our previous analysis we made several suppositions which were not at first obvious. Before any further search for applications of the obtained formula (23), it seems necessary to consider these suppositions in some detail.
3. SUPPOSITIONS
When deriving the formula (23) we assumed the validity of (3)-(4) as well as the inequalities N2 D 1, 8’ << 1 and a@ CCa,. Inequality (3) for the case of whistler propagation@ < Qisvalidfor all nifit is valid for n = 0 and n = 1. Hence after substituting (23) into (3)-(4) we can present them in the form of a system of three inequalities :
&&o”/(n”(n-
coy) <<1
&&B2/Q2 <<1 &JLQu2(Q2 - 034,,/2n4
(26) (27)
<< 1
where A,, = w:/wi = A,/(j+ 1). When deriving (26)-(28) we took into account
(28) that
k;i N N2c04/c2Q2 k: u N2c02(C12 - 02)/S12c2.
To estimate the domain of the validity of(26)-(28) we can take the largest value of l&l equal to m. Substituting (24) into (26)-(28) we can join these three inequalities into a single one :
4
<
(29)
where Icl =
Jlaol(tko)2JiF7/(2JZw3)
K* =
JlaolJZG7/(2Jzo)
Kj = Jlaol.
a0
0
-1 FIG. 1. PLOTS
a,
VS
o/n
FOR
A, = 1
(DASHED).
(SOLID)
AND
A, = 2
CP(j + l)/(JZA,wJZZ7).
Plots K1,2,3 are presented in Fig. 2 for A, = 1 (solid) and A, = 2 (dashed). j is supposed to be equal to zero ; generalization of our analysis for j # 0 is straightforward. From Fig. 2 it follows that (29) is never satisfied for w z 0.69R (A, = 1) or w = 0.65R (A, = 2), that is for those frequencies when a, = 0. Outside the vicinity of these frequencies the range of w for which (29) is valid strongly depends on A,. When A, = 1 and w < 0.69R (29) is satisfied for larger fl than in the case when A, = 2 in the same frequency range and vice versa for w > 0.69Q. Keeping in mind that for the real magnetospheric conditions /3 is in most cases (with the exceptions of magnetosheath and plasma sheet conditions) less than 0.02, we can expect (26)-(28) to be valid for A, = 1 and w 5 0.6a.
Quasiel~trostatic Ki
K3
KI
K3
299
whistier-mode propagation 1
“3
FIG. 3.
PLOTS
tlo,ke
VS W/a
FOR z‘f, =
1 (SOLID)
AND
A,
= 2
(DASHED).
FIG. 2. PLOT‘sicf,Z,S VS co/ii
FOX
A, = f (SOLID)AND
A,
=
2
(DASHED).
o i= 0.6R. In order to assess them for a particular case one can use the curves plotted in Figs. 2 and 3. Let us consider inequality N2 >> 1. When considering (23)-(25) we can obtain the expression for the maximum value of N2 (which is attained at B = BRwhen a,
n2/Q2.
(30)
Taking into account that l&,1> 1; la,( P /I (or laol >>/S (when w is not small) and II* >>f2’ we can see from the latter formula that N&, >> I. From (25) we can see that this inequality is satisfied not only for N2 = Ni, but also for other values of N2 if B is not too far from 8,: 10’15 lo--t?,l << 1. The last inequality was also supposed to be valid when deriving (23). In particular, it was explicitly used when we neglected the terms proportional to B’@in the derivation of (23). Strictly speaking, when making the last operation we required not If71c 1 but it71CClao/ael. In order to control the latter inequality we plotted in Fig. 3 the curves [a,/~,1 vs w/Q for A, = 1 (solid) and A, = 2 (dashed), As follows from this figure the restriction on 0 in order that our model could be valid, is less stringent for A, = 1than for A, = 2. In the vicinity of the points where ae = 0 our model also is violated because of the restriction on 0’. Thus it follows from the analysis of this section that the suppositions used when deriving the formula for whistler refractive index in the form (23) are not contradictory and can be simultaneousIy satisfied for realistic magnetospheric conditions, at least for
4. DISCUSSION
As was already pointed out in Section 3, the value of NZ attains its largest value in the vicinity of 8 = 19,for aoi:OorB=B,a for a. > 0 determined by (30), contrary to the case ofwhistler-mode propagation in a cold plasma where N2 + 03 when B -+ 8,. Let us now present an expression for NiBX in a more convenient form :
As follows from the latter form&a Ni, --+co when wii -+ 0. K&X also becomes infinitely large when a, -+ 0 but in this case our model is no longer valid. It can be easily proved that in the whole range of 8 considered, N2 is a monotonically increasing function of 8. We now consider the problem of wave energy propagation when based on the derived formula (23). However before proceeding in this direction, we recall some fundamental points of the theory of wave propagation (Budden, 1961; Six, 1962 ; Helliwell, 196.5).We presented in Fig. 4 a schematic plot of a part of the curve N(B) in polar coordinates (solid segment) for 0 close to 19,.Axis N,, as well as the dashed segment on the same figure coincides with the direction of the external magnetic field B,,. A perpendicular to the curve
S.S.SAZHIN and E. M.
300
VOh /
N, FIG.~. SCHEMATIC PRESENTATION 0~ em DIRECTIONS V, AND v, CORRE.WONDlNG TO THE SEGMENT OF THE CURVE M(e) (SOLID).
N(O) at the point corresponding to 8 = 8, indicates the direction ofwave group velocity which forms an angle $ with the magnetic field. An angle between V, and V,, we labelled, following Stix (1962), 8,; it is determined by the equation (Stix, 1962) : Jtan @,I= IN-r*dAJ/dBl.
(32)
After substituting (23) into (32) we have : (tan @,I= (2Jm)
- ‘.
(33)
Let us first consider the case when a0 > 0. From (33) it follows that \tan 8,1-+ cc when I#i -+ @ which implies that [es] -+ 7r/2for these B’,similarly to whistlermode propagation in a cold plasma. The value of II/($,,) in this case is determined by the formula : $c, = n/2-8,tJ2a,P.
(34)
The value of #, determined by (34) is larger than the corresponding value of $, for whistler-mode propagation in a cold plasma at 6 close to ORwhen $,, YE$, = R/2-&. For a, < 0 it follows from (33) that ltan @,I4 l/ ,,/?$a when 8’ -+ 0. Hence the value of $, is determined by the equation : $,, = larc~an(UJGiiiF)--~~l.
(35)
The value of $, determined by (35) is obviously less than $,. The case a, < 0 corresponding to o/Q < 0.65 when A, = 2 (or 0.69 when A, = 1) seems to be more typical for whistler-mode waves observed in the magnetosphere (see Sazhin, 1982a). it will further be of our primary attention. The determination of the values of $ in the vicinity of B = e, is especially important for several magnetospheric applications. We will consider briefly one of
SAZHINA
them. The value of $,, was used when determining the height of generation of saucer emissions and V-shaped hiss registered by satellites in the topside ionosphere (Mosier and Gurnett, 1969). When making their estimates these authors used a cold plasmamodel and it isnot obviousapriorithat thismodelisvalidfor thereal magnetospheric conditions. In order to check .its validity let us calculate $,, when using (35). Supposing that II/C2 = 3 ;mewi/ = 1 eV we have: /? = 1.8 lo- ‘. Taking further o = 0.4Q A, = 1, and, correspondingly a, = -0.44, we have arctan(l/~) = 89.5”. Hence it seems justified for us to use the methods suggested by Mosier and Gurnett (1969) when determining the height of the region of the considered emissions generation within the cold plasma model. At the same time the influence of the electrons’ finite temperature and anisotropy on the value of $,, seems to be much more important in the equatorial magnetosphere. Let us consider for example plasmaspheric conditions: II/Q = 100; mew;/2= 1 eV; /I = 0.02. Taking again o = 0.44 A, = 1 and correspondingly laoI = 0.44, we have: arctan (l/s) = 75.1”. Hence $, determined by (35) is equal to X.7”,which is less than the corresponding value of $C = 23.6”. Our analysis of Section 3 made it possible to justify the validity of our formula(23) for wave normal angles 0 close to 6a (or 0,-a). In order to control its validity for B away from ORwe presented in Fig. 5 the curves N(B) calculated from the formula (23) (solid), from the equation (13) (dotted), and from equation (14) where no restrictions on 0 were imposed (dashed), when supposing that w = 0.4!& II = lOR, mew!/2 = 10 eV (,Q= 0.002). Parts of these curves are presented in Fig. 6, when using another scale of N,, axis. As follows from these curves the plots calculated from (23) and (13) practically coincide when B 2 38,/4. At the same time for B < 30,/4 the curve calculated from (13) practically coincides with the corresponding curve calculated from the formula derived by Sazhin and Sazhina (1982) [see their formula (25)] even within the accuracy of Fig. 6. Thus our formula (23) can be considered as complementary to that which has been derived by Sazhin and Sazhina (1982). Since the value of N calculated from the latter formula tends to decrease when 6 approaches O,, we can expect that for a certain 0 = 0, the curves calculated from (23) and from the formula of Sazhin and Sazhina (1982) intersect. Thus when modelling whistler-mode propagation in the real plasma we can set N(0) =
N,(e) N,(e)
0G 02
0, e,
(36)
where N r(@is determined by formula(25) of Sazhin and
301
Quasielectrostatic whistler-mode propagation
FIG. 5. PLOTS N(o) CALCULATED FROM THE FORMULA (23) (SOLID),FORMULA (13) (DOTTED), FOKMULA (14) (DASHED);DOTTED CURVE c01~clDEs WITHIN THE ACCURACY 0~ A PLOT, WITH THE CORRESPONDING cuwE CALCULATEDPROMFORMULA (25) OR SAZHINAND SAZHINA (1982).
Sazhina (1982), N,(H) is determined by formula (23) of our paper. The complementary character of the curves N,(O) also occurs for A, = 2 and other values of o/n and /I. it is important to notice that the curve calculated from (14) which at first sight better approximates the value of i?(n), appears to be slightly worse than that which was
calculated from (23). [Error which comes from presenting equation (13) in the form (14) seems to be partly compensated by the error coming from the condition 0’ c< 1.1 Hence it seems sufficient to use (23) when approximating N(B) even if f3 is not very close to 8,. 5.
(a) We have derived a formula for N(B) for whistlermode propagation in a hot anisotropic plasma valid when 8 is close to 0,. This can be considered as complementary to the corresponding formula for quasilongitudinal whistler-mode propagation which was derived by Sazhin and Sazhina (1982). (b) The derived formula describes whistler-mode propagation similar to that for a cold plasma for 0 < OR but, contrary to the case of whistler-mode propagation in a cold plasma, it predicts finite N when B approaches B,. (c) From the derived formula, it follows that the electrons’ finite temperature and anisotropy influence the direction of V, when 6 is close to 0, ; this influence is especially important for the equatorial regions of the magnetosphere where the value of /I is not so small (- 10m2) compared for instance with its value in the aurora1 topside ionosphere (- IO-‘).
t
4
---%_____Nc'
I 10
I
REFERENCES
50
NJ FIG.~.
CONCLUSIONS
THEPARTSOFTHE PLOTS PRESENTEDON ANOT~SRSCALEOF~ ,, AXIS.
FIG.~ WITH
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S. S. SAZHIN and
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