Download full-wave vlf modes in a cylindrically symmetric enhancement of

Planet. S~~nct?
.ScL. Vol. 23. pp. 1649 to 1657. Pergamon Press, 1975. Printed in Northern Ireland
FULL-WAVE VLF MODES IN A CYLINDRICALLY
SYMMETRIC ENHANCEMENT
OF PLASMA DENSITY
M. J. LAIRD and D. NUNN*
Dept. of Mathematics, Ring’s College, London WC2, England
(Receiued 21 April 1975)
Abstract-VLF
field. It is required to account for the good transmission
properties of these ducts as some leakage of energy is expected, with consequent attenuation.
We consider a cylindrical duct, whose axis is parallel to a uniform magnetic field. The electron density
of a cold plasma is a function only of radial distance r from the axis, taking constant values in inner
and outer regions r < a and r > b, and varying smoothly in the duct wall a < r < b. We compute
full-wave ‘trapped’ VLF modes. Results are presented for a range of values of the parameters of the
model. In general we find that attenuation of leading modes is very low, agreeing with observation.
Specific features are that leakage (arising from mode conversion in the duct wall) increases as the wall
is made thinner, that leakage for the leading modes is less for weaker ducts, that many modes may
propagate, and that, because of interference effects, some higher order modes also suffer little attenuation. _
1.INTRODUCTION
It has long been known that VLF waves in the
magnetosphere may be channelled along magnetic
field lines by means of ducts consisting of field
aligned enhancements of plasma density of a few
percent. The refractive index for waves propagating
in the whistler mode varies with plasma density, and
in consequence the ducts act in a similar way to
dielectric waveguides. Their transmission properties
are known to be good, since VLF radiation from
lightning strikes can travel to the conjugate point
without difficulty-a
path of about 60,000 km for
a field line with an L parameter of four. The ducts
are believed to be generally of circular or elliptical
cross-section, and about 10-20 km in diameter at
ionospheric heights implying dimensions of a few
times this deep in the magnetosphere.
Evidence
for ducting,
together
with full
details of and reference to work done prior to 1965
is given in Helliwell’s book (Helliwell, 1965). More
recently, Angerami (1970) has discussed observations from the OGO 3 satellite. On the theoretical
side, Walker (1971, 1972) and Scarabucci and Smith
(1971) have treated in considerable detail models
with planar geometry, using both phase-integral
techniques and full wave methods. These studies
showed that indeed duct walls are extremely good
reflectors of wave energy for small angles of incidence. Cylindrical geometries have been studied by
* Now at the Royal Aircraft Establishment, Farnborough.
Klozenberg et al. (1965) and Boswell (1970), with
metal or vacuum as an outer boundary.
In this
paper we present results of full wave calculations
for a model with cylindrical geometry, the plasma
being unbounded. A major aim has been to find the
amount of attenuation arising from mode conversion
in the duct walls.
2. THE MODEL
We take a cold electron plasma, all ions being
assumed to be immobile. At the frequencies we shall
be considering, the magnetospheric plasma may be
considered collisionless (lossless). However, it is
often convenient to have a damping term, so
provision is made for its incIusion.
Our model duct is then as follows. The duct axis
is taken as the z-axis in a cylindrical coordinate
system (r, 4, z). Parallel to this axis is a uniform
magnetic induction B,. The electron density is a
function, N(r), only of distance r from the axis. It
is a constant, N,, in an inner core r < a, a constant,
Nr,, in an outer region r > b, and varies smoothly
from iV, to IVYwithin the duct wall a < r < b. (Note :
in the rest of this paper subscripts a and b will be
used to denote quantities in the inner core and outer
region respectively.) As the equations of the problem
have to be integrated numerically for r between a and
b, virtually any variation for N(r) may be used. In
our computations we chose a sinusoidal variation
with half-period (b - a), giving a maximum density
gradient half way through the duct wall.
1649
M. J. LAIRDand D. NUNN
1650
3. MATHEMATICAL DEVELOPMENT OF
THE FIELD EQUATIONS
Hence
A comprehensive development of the field equations for a general cylindrical geometry is given by
Allis et al. (1963). Accordingly here we restrict our
treatment to what is necessary for the problem in
hand.
We suppose that all components of the wave
electric field E and magnetic induction B are given
by the real parts of complex quantities of the form
F(r) exp (i5), where
5 = kz + m# - wt,
(1)
m is an integer, and k and w are constants. In our
analysis we shall take w to be real and k complex,
the imaginary part of k being of particular interest.
The linearised equation of motion of the plasma
electrons gives, in SI units,
iwm&j = -N(r)&
-I- ej x B,,
(2)
where e and m, are
the electronic charge
magnitude and mass, and j is the current density. In
the absence of collisions E = 1; otherwise E =
1 + iZ, where Z is the ratio of the frequency of
collision of electrons with ions to the angular wave
frequency w. Now
curl B = pc,j - iwE/c2.
(3)
r a(rm,)
;iF ~;i;+ (q2 - ka)r2Be - m2B, = 0,
(10)
and so
B, = V,(2r)erc,
where
R = q2 - k2,
(I 1)
and Fm(llr) denotes a cylindrical function of order
m, i.e. a linear combination of the Hankel functions
H’1*2’(lr). The remaining components of B and E
rn:y then be calculated from equations (9) and (5)
respectively.
In the duct wall, N is varying, the above analysis
cannot be used, and numerical integration must be
resorted to. We return to equations (5) and (6), and
note that B, and E, are nowhere differentiated with
respect to r. These may then be eliminated, and
there remain four coupled first order ordinary
diffe~ntial
equations which may be integrated
numerically:
dFa/dr = TirF3, i, j = 1, 2, 3, 4,
(12)
where the repeated subscript is summed over, and
(ES, cB,, E+, cB+) = (F,, F2, Fe, FJeSi.
The elements of the matrix (r,,) are functions
alone, and are given here for reference:
(13)
of r
Putting
X = Ne2]e,m,w2,
Y = eB,lm,w,
.(lr)
comparison of equations (2) and (3) indicates that
we may omit the displacement current term in (3) if
X > 1, i.e. if the wave frequency is we11 below the
electron plasma frequency. With this omission on
substituting for j, equation (2) becomes
5cs curl B = ioxE
+ iYc22 x curl 3,
(5)
where E is the unit vector B&Z,.
To develop solutions for the regions where N is
constant, we note, following, e.g. Klozenberg et al.
(1965) and Boswell (1970), that equation (5), together with
curl E = iwB,
(6)
curl B = qB
(7)
is satisfied by
provided q is a root of
T,, = T,, = Td2 = 0,
T,, = -T,,
= -km/Arc,
A = iwXjc2(1 - Y2)
T13 = Tza = -k Y,
T,, = (ka - iwA)/Ac,
T,, = -Td3 = ikmclwr,
T,, = -T,
-z--l
(14)
= my/r,
T,, = -(k2c2 -I- Xw2)i/wc,
T,, = (ioAr
- ~)[Ar2~,
Tdl = (m2c2 + w2r2X)z/wr2c,
Tu = --l/r.
Note: the above matrix elements are given for the
case of no collisional damping.
If collisions are
inchrded, X and Y shouId be replaced by X/g and
Y/E, respectively.
4. TRAPPED MODES
{q* - kYq + w2X/c2 = 0.
(8)
Equation (7) occurs in a number of contexts, and
solutions are well known.
With the assumed
dependence on 4 and z,
qrB, = imB, - ikrB+,
‘I”B+ = -kmB,/r
- q aS,l ar.
(9)
We now look for ‘trapped’ modes of the system;
i.e. we try to find solutions of the field equations
satisfying appropriate boundary conditions for which
the fields do not extend appreciably outside the duct.
Before giving a detailed presentation of the results
it is useful to have a general picture of what to expect.
VLF modes in cylindrical duct
If we put k = q cos 8, equation (8) yields the
familiar dispersion relation for a whistler propagating at an angle 0 to B,, namely (assuming no
collisions)
qa = dX/c2( Ycos 8 - 1).
Equation
(15)
(8) may then be rewritten as
q8 - kYq + kp2( Y - 1) = 0,
(16)
where k, is the wave number appropriate to a
whistler propagating parallel to B,, i.e., Y > 1, and
kDa = 02X/C2(Y - 1).
(17)
Figure 1 gives a plot of q/k against (k,/k)2 for given
real k: a parabola with axis q/k = Y/2. When
k, = k, the two roots of equation (16) are ql = k,
and q2 = k(Y - 1). For Y > 2, we identify q1 as
the smaller of the two roots, and note that t, (=
(VI2 - k2)li2) is real for k, > k and imaginary for
k, < k. On the other hand, for 1 < Y < 2, we
identify q1 as the greater root and the conditions for
A, to be real or imaginary are reversed.
In an ordinary dielectric cylinder, for the guided
modes the argument of the Hankel functions occurring in the solution is real in the dielectric itself and
imaginary outside, so that the field amplitudes decay
exponentially with distance from the axis (see, e.g.
Collin, 1960). Similarly here we look for a solution
for which 1, is real in the core and imaginary in the
1651
outer region. The analysis in the previous paragraph
shows that, for small changes in density, this may
occur near q = k = k,. Specifically, for Y > 2, we
need k,, > k > kpb, so that the density must be
enhanced in the duct, whereas for 1 < Y < 2 we
need kgb B k > k,,, that is, the density within the
duct must be diminished. (k, a W2.)
Ray theory
(Helliwell, 1965) leads to a similar conclusion. Thus
we expect a wave, which we shall refer to as the
principal wave, for which propagation in the core is
nearly parallel to the duct axis and reflection occurs
in the duct wall.
There is, however, a complication in that another
wave exists for which the associated root q2 is near
to k( Y - l), so that As2is approximately given by
k2 Y( Y - 2). For Y > 2, A, may be real everywhere,
and so the wave, the escaping wave, may carry
energy away from the duct. Now in the duct wall,
the varying of the refractive index leads to mode
conversion. Part of the principal wave is converted
into the escaping wave so that attenuation may be
expected, giving complex values for k. But from the
work of Scarabucci and Smith (1971) only a small
amount of conversion is likely, so that Im(k) <
Re(k), and the above discussion in which k is taken
to be real corresponds closely to the actual situation.
When Y < 2, 1, may be imaginary everywhere,
and so solutions may be expected for which no
energy is carried away from the duct and hence there
is no attenuation in the direction of the axis.
Klozenberg et al. (1965) noted that the modes for
positive and negative values of m were different. This
\
I
(I,Y-1:
ik
Y
(191)
e
b
I
t1,1:
Y
?
1
(I,Y-II
0
I<
Y>2
-
Y<2
(/$/Xl*
FIG. 1. WAVE NUMBER
AS A FUNCTION
OFk,%FORGIVENk.
k,,”is proportional to the electron number density. Real
values for I corresnond to a/k > 1.
lack of symmetry can be seen from expressions (9) for
B, and B+. In an isotropic medium, q is plus or minus
the angular wave number, the signs corresponding to
right- and left-handed polarization respectively, so that
symmetry may be maintained by changing the sign of
both m and q. In our situation, this is not possible: the
two roots of equation (16) have the same sign. This
corresponds to the fact that the polarization of whistler
mode waves is determined by the uniform magnetic field.
For an observer looking in the direction of this field, the
electric and magnetic vectors for the wave rotate in a
clockwise, or right-handed, sense.
When examining our results, we noticed a considerable
similarity for modes with m = 1 & M, i.e. near symmetry
about m = 1 rather than m = 0. Using recurrence
relations for Bessel functions, equations (9) and (11)
give, for m = 1 i M,
B, =
$ +?&r)
Bb = - % if,(h)
i(M f 1)
WmdW,
r(k + q)
Wf i 1)
+ VxdW,
r(k + q)
+ ~
B. = f %?x*~&).
(18)
(The factor exp (it) is understood.)
For trapped
modes, solutions have to be matched at the duct walls,
M. J. LAW
1652
and D. NUNN
and here the ratio of the second term to the first in the
expressions for both BP and B+ is of order A/ak5. For
the principai wave, this ratio may be expected to be
very small, as t <k, and for typical whistler wave
lengths of a few kilometres a duct diameter of order
100 km means that ku is quite large. B, is small compared
with the transverse field. For the escaping wave, il and k
are of the same order, so the first term in the expressions
for B, and B+ is still dominant, while i(a is quite large,
and VX+&Aa) and %?,,&&z) are approximately 180”
out of phase. Thus, except for the centre of the inner
core, the fields (apart from the angular dependence) for
m = 1 f M may be expected to be much the same. In
fact, the difference in angular dependence is largely
geometrical. The principal wave propagates nearly
parallel to the z-axis, and is close to being circularly
polarized. (B+ $ iB?.) Transformation to a Cartesian
coordinate system is thus almost equivalent to muhiplication by exp (-i#), changing the angular dependence
to exp (iiM+). The fundamental mode may be. associated with m = 1 rather than with m = 0, the magnetic
induction of the principal wave being given approximately by B, = -iB,. = (ik/il)g,,,(tr) exp i(kz - of).
inner boundary condition to be satisfied is that of
finite fields at r = 0. In the core, the above expression for B, must reduce to a sum of Bessel functions
of the first kind, so that a,, = a,$ them. If damping
is inserted in the outer zone, the outer boundary
condition is simply that all fields go to zero as
r -, to. An alternative procedure is to put zero
damping in the outer region and select two waves
only-the
principal wave (here non-propagating
that goes to zero as r -+ co, and the escaping wave
whose Poynting vector is directed outwards. The
procedures are almost equivalent provided that the
degree of damping in the outer zone is neither too
large nor too small, and give that two of the four
coefficients at5 must vanish in the outer region.
We now consider two solutions in the inner core of
the form
Trapped modes may also exist when Y > 2 for a
dim~ution in density in the duct (a trough). As can
be seen from Fig. 1, as kp2, which is proportional
to
electron density, is increased, the two roots for q
become equal, when q/k = set 0 ==,X/2, and then
complex. Suppose that
B, = H’2’(A2r)
+ Rg2HE’(A2r) + R&‘2’(Q)
vn
9n
k2 < kpa2 < k2 P/4( Y - 1) < kpb2.
(1%
Then R, and rZ, are real in the core and complex in
the outer zone, so that waveguidelike modes should
exist, with solutions decaying exponentially with r
outside the duct. One can no longer distinguish
between principal and escaping waves. Both waves
propagate in the core; for one wave t?is greater than
COS-~(2/Y), and for the other it is less. (If the
inequalities (19) hold, for some value of r in the duct
wall the density is such that 0 = co@ (2/Y), and
here the associated ray direction is parallel to the
duct axis (Helliwell, 1965).) For propagation at an
angle to the magnetic fieId, Landau damping is
expected to be important.
(We return to this point
in our concluding remarks.) Accordingly, though
these modes have interesting properties, we shall for
the most part discuss propagation in densityenhancements.
5. APPLICATION
OF BOUNDARY
CONDITIONS
In regions where the electron number density is
constant, for given m, o and k the general solution
for B, is, from equations (8) and (1 l),
6~2
3=2
(20)
the factor
exp (i4) again being understood.
The
B, = H”‘(Q)
m
+ R,H’2’(~r)
m
+ &H”‘(12r)
m
’ (21)
’ (22)
where the complex coefficients Rll, Rle, Rzl, Rzz, are
calculated by integrating equations (12) through the
duct wall and imposing the outer boundary conditions. The two solutions correspond to outgoing
principal and escaping waves respectively, together
with the reflected waves, (R,& forming a reflection
matrix at the inner core boundary r = a. Taking a
linear combination
of these two solutions that
satisfies the condition of finite fields at the origin
gives the condition for trapped modes in the form
C = 1 - R,, - R12R&1 - R2,)-l = 0.
(23)
The problem thus becomes one of finding zeros of C
regarded as a function of k.
As we were interested in finding propagating
modes with low attenuation,
the procedure we
followed was to take k = k,,(l - nA), (0 < A < l),
and compute C for successive values of n (= 0, 1,
2 **, for Y > 2, = 0, -1, -2, . . . , for 1 < Y <
2)‘: On finding a minimum of [Cl, the nearby zero of
C was then computed. The procedure was checked
to see that the attenuation factor Im(k) so obtained
was independent of the number of steps used in
integrating numerically the equations (12) from
r i= a to r = b, and was also independent of E when
5 + 1. Another check made was that the radial
components
of the Poynting vector added up
correctly when k was real.
Typically, we found that for propagation well
away from half the eiectron gyrofrequency (corresponding to Y = 2) IRlll would be very close to unity
for k > k,$, and modes existed. As soon as k
VLF modes in cylindrical duct
reached kpb, i.e. the principal wave became propagating in the outer region, 1&J would fall rapidly,
and modes ceased to exist.
For propagation at frequencies just below the
half gyrofrequency, that is, with Y slightly greater
than 2, the situation is more complicated. As k is
decreased in the search procedure, it is evident from
Fig. 1 that the two roots of equation (16) get very
close. It was found that modes can exist for which
l&l is not close to unity, indicating a considerable
amount of coupling in the duct walls between the
principal and the escaping waves.
For troughs with Y > 2, &a and Rsl are essentially
of unit magnitude, i.e. as one follows a wave through
a number of reflections, the propagation angle 0
alternates between its two possible values in the core.
6. RESULTS
The principal difficulty encountered in presenting
the results is that even for our simple model there are
a number of parameters and to give full results for
even a few values of each would take up much space.
In addition, the density profile for a < r < b is
arbitrary.
For all our computations, unless stated
otherwise, we took a sinusoidal profile, with
2N(r) = N, -t ND + (N, - NJ sin [n(a + b - 2r)/
2(b - 41 (24)
1653
within the duct wall. Accordingly, we first present
some results appropriate to conditions at the equator
in the region of L = 3, and then vary one or two
parameters at a time to indicate what happens, for
example, as the duct wall is made thinner, or the
density enhancement
reduced.
Initially we take
a = 40 km, b = 60 km, w/27r = f = 12 kHz, and
electron plasma frequencies in the inner core and
outer region of 1Of and 9.75f respectively. This
corresponds to a plasma density enhancement of
about 5 ‘A over a cylinder about 100 km in diameter.
We suppose that propagation is at a little below half
the electron gyrofrequency, with Y = 2.2. Table 1
lists modes for different values of m, giving for each
mode the decay length l/Im(k) in megametres, and
the real parts of a& and a&. For comparison,
&LX = 91.8. The near symmetry about m = 1
shows up very clearly. The results for m = 0 and
m = 2 were found to be almost identical, while even
for m = -18 and m = 20 the values for L are very
similar, though there are differences in decay length.
There is a close correspondence between the values
of Re(a&) (the imaginary part is very small) and
the zeros of J&x), where m = 1 f M, the latter
being about 10 % greater than the former. This compares with the asymptotic results of Snyder (1969) for
a guide consisting of a dielectric cylinder embedded
in a medium with slightly differing dielectric
TABLE 1. RESULTS FOR A 5% ENHANCEMENT OF PLASMA DENSITY
OVER A CYLINDER ABOUT 1OOkm IN DIAMETER. PROPAGATION
ISATJUSTBELOWHALPTHEELECTRONGYROFREQUENCY(Y=
2.2)
ID-O,Zl-2
l/Idk)
ow
(6)2:0
2.18
m.aj
(5)1.3
(4)l.Y
5.00
1.84
60.62
60.25
(3)5.4
(3)3.1
0)l.O
(3)8.4
(2)6.9
(JD.6
(2)3.8
(2)3.0
cu7.3
w2.9
(2)j.O
10.60
13.51
16.33
19.14
21.94
24.73
27.49
30,28
33.01
35&n
%.I7
59.71
59.01
58.15
57.12
55.93
54.57
53.04
51.32
49.45
47.32
44.82
l/Idk)
0-M
(3)l.J
(2)4.8
(2)1.3
Rebh&
R&$J
22.14
26.20
29.76
55.84
53.18
51.66
(3)6.1
(2)l.O
33.14
36.50
49.36
46.76
R4a “l&J
(4)5.5
(4)Z.l
(4)Z.l
(6)l.C
3.41
6.36
9.22
12.06
(3)6.3
c3n.4
(3)z.l
(3)3.3
(2)3.0
(4)6.2
w.3
(2)1.8
14.89
17.71
20.52
(2)9.7
R4ai\*J
60.75
60.46
60.01
59.39
58.61
51.66
23.32
26.10
28.87
56.55
55.28
53.83
52.21
31.65
34.37
37.26
50.41
48.44
46.13
22.10
26.15
29.71
33.09
36.44
55.05
53.6
51.69
49.39
46.81
m-_18
ow
(3D.8
(2)3.7
(2b.5
(5D.3
(06.9
1654
M. J. LAIRD and D. NLEW
constant. Another feature is that for given m the
attenuation
does not always increase with mode
number as one might expect. One may think of an
outgoing principal wave in the core as being reflected
by the duct wall, with a small part being converted in
the wall to outgoing and incoming escaping waves.
This incoming wave is then reflected at the origin and
propagates straight out through the wall (R,, is
very small). The two escaping waves in the outer
region may destructively interfere if the phasing is
right, and this may prevent energy from escaping
from the duct. Some high order modes may then be
virtually unattenuated.
A sketch of the situation is
shown in Fig, 2. Figure 3 shows a pfot of Im(E+)
(for 5 = 0) for the first two modes associated with
m i= 0. The effect of a small amount of escaping
wave added to the principal wave is clearly seen in
the short period oscillations.
The results indicate generaIly very low attenuation
rates. For many modes, decay distances are very
long compared
with a typical magnetospheric
distance of order 50 Mm. Thus a large number of
modes might be involved in ducted propagation. For
given m, the modes cease as the values of aA,, and
u#& approach each other so that modes may be
expected up to about m = 130. (The lowest zero
of Xx w M -i- I.856 &P/3‘)
We now examine what happens as the amount of
enhancement, the duct size, and the wave frequency
are changed. We consider propagation at below
no.
3. AZIMUTHAL
COMPONENT
OF BLECTRIC FIELD FOR
m = 0.
In the core (r < 40 km) the field component varies
approximately as .&(&J; the small oscillations show
the presence of the escaping wave.
haIf the electron gyrofrequency, i.e. Y > 2. As ic
is decreased in the search procedure, the values of
A,, and $, approach each other, becoming equal
when q = ICY/~, i.e. when
k2 = 4k,,a2( Y - I)/ F,
(25)
giving
aI,” = d,,s = kp,2( Y - l)( P - 4)/ y2.
(26)
For lower values of k, A, and A, are complex
conjugates, the waves in the core are i~omogeneous,
and one would not expect trapped modes to exist
even if there were no mode conversion. Expression
(26) increases with Y (note that kpa2(Y - I)
depends only on NJ so that initially as the frequency
is decreased from half the electron gyrof~quency the
number of modes for given m wiil rise. For large
enough Y, the value for k given by equation (25) will
be less than kDa, so that modes will now cease when
k = &,,a, as the principal wave then becomes
propagating in the outer region. The associated
vaiue of iii, may be found from equation (16),
putting k9 = kpQ, k = kpb, and q2 = k2 i- ii2, and
may be shown to decrease with Y increasing. The
maximum number of modes may then be expected
to occur when
ks = kpb2 = 4kss2(Y - I)/ y2,
WI
Y = 2/(1 - @‘a),
(28)
giving
where
6 = (k,,2 - k,,,2)jk,,2 = (N, - N&/N,.
(2%
Substituting for Y in equation (26) gives as the
maximum value of AI, for which trapped modes may
be expected to exist
-
ey,
A~, = a,, = k,,(Y
Destructive interference in the outer zone may reduce the
which varies as the square root of N, and as approximately the fourth root of 6, which measures the
OF THE PRINCXPA?~ WAVE (1) AND
CONVEXSXON TO THE ESCAPlNG WAVE (2) IN THE DUCT WALL.
amplitude
of the escaping wave and the energy loss.
-
1yw4(2
FIG, 2. IWLECTION
(30)
1655
VLF modes in cylindrical duct
TABLE 2. VARIATIONOPTHENUMBEROFMODESWITHFREQUENCY
m_lu
f9)Y.O
(6)E.l
(6)l.l
(514.4
(514.3
(5j7.2
(SE.%
(5)Z.B
(5)6.%
(6)Z.O
f5I2.4
(4)5.%
(412.2
(413.8
w3.7
OI4.6
OD.6
(414.4
(2f3.1
121-l.%
{l&l
Oh5
fl16.5
&a%
7.64
IO.38
13.11
15.79
18.43
21.02
23.36
26.00
28.42
jo.aX
33.17
35.50
37.81
40.11
42.39
44.68
46.97
49.27
51.62
53.99
56.45
59.16
95.23
94.90
94.44
93.84
93.12
92.29
91.35
90.33
89.22
Bs.03
ij9)z.o
2.02
(m.5
4.63
(7)6.%
7.22
t7)l.S
(614.3
9.76
12.21
(5I5.5
(4j6.5
(3I6.8
14.53
16.70
18.72
(214.3
20.53
177.09
176.74
176.13
175.27
174.21
172.99
171.6%
170.31
168.97
yL
==_&
86.17
05.43
84.02
l/b(k)
b)
=&.‘y,)
Reb?-2a)
82.54
81.00
(7j4.1
(G4.4
1.93
4.40
278.22
277.20
79.3%
77.69
75.93
74.07
72.12
7o.Ob
(5)6.1
(4X.5
w4.5
6.81
9.10
11.14
275.64
273.47
271.02
67.05
65.31
amount of enhancement. For a given duct structure,
the values of aA&,for modes depend on the frequency
(i.e. on Y). However, in our computations
the
dependence was rather weak, and the maximum
number of modes did occur for Yin the region of the
value given by equation (28).
The effects of changing the wave frequency,
keeping a, 6, the electron plasma frequencies and
the electron gyrofrequency unchanged, are shown in
Table 2. S = 79/1600, and equations (28) and (30)
give for maximum modes values of 2.57 and 63.2 for
Y and a& respectively. Note that for Y = 254
the number of modes has increased to 24 (compared
with 14 for Y = 2*2), while it has dropped to 5 for
Y = 8.8. (m = I in each case).
Table 3 gives results for a smaller duct, with a and
The reduced
number of modes is immediately apparent, though
the attenuation for the leading modes is still low.
Vatues for m = -1 and m = 3 are given to show
the sy~et~
about m = 1.
In Table 4, the effect is shown of weakening the
duct. We now take electron plasma frequencies of
120 kHz in the core and 119.4 kHz in the outer
region, corresponding to an enhancement of 1%.
Equation (28) gives Y = 2.22 for the maximum
mnnber of modes, so the results shown (Y = 2.2)
are close to this maximum. One point to notice here
is that the attenuation is less than for the stronger
duct, presumably because the density gradients in
b equal to 5 and 25 km, respectively.
TABLE 3. RESULTSFOR ASMALLDUCT
(Y=
2.2)
&
l/-(k)
Rebhla)
Rdahza)
(5b.8
(411.9
1.31
1.47
2.92
4.41
6.91
5.97
oa
il)M
lb(k)
NJ
(3j6.3
(lJ3.4
4.37
6.~0
lb(k)
bw
Re(ahlJ
R&h&J
(412.3
(114.3
2.76
4.35
6.96
6.01
16.56
M. J. LAIRD and D. NUNN
TABLE4. RESULTS
roa A WEAKDUCT(Y = 2.2)
m_l
l/h(k)
(K-n)
Reb?-la) Re(a\a)
J&(k)
w
WayJ
R~b~,f
(Bj7.Q
2.09
a.03
j4D.5
22.80
55.53
(7h.3
A.79
60.64
(412.5
25.16
54.34
(6)Z.l.
7.49
60.30
(4&.3
27.51
53.04
WI.4
29.83
51.61
(5jA.l
10.17
59.82
(S)LY
1".02
59.20
W5.6
32.17
5w??
(:)I.‘!
:..A1
5fGi
(3h.7
34.51
48.33
(4J5.2
i4ir.e
17.94
2,,.AO
57.58
%.M
m.4
36.36
46.36
TABLE 5. RESULTSFORATHIN-WALLED
(5% ENHANCEMENT, Y=
2.2)
DUCT
m.l
l/-k>
bsn)
(3j2.5
(2)3.3
(2j1.4
w-3
(2)&O
(2k.5
(lj2.0
(20.3
Reb$,)
ah&a)
%‘I&)
WJ)
Reb%Ia) R&bza)
2.32
5.33
8.38
74.53
74.34
73.99
(04.5
W3.7
(04.1
26.47
29.81
11.44
73.48
14.47
17.43
20.50
23.63
71.06
w2.2
HALFTHEELECTRONGYROEREQUENCY
(Y = 1.91)
67.10
(2)1.4
32.50
35.87
63.65
72.83
(2f1.4
41.59
59.73
72.04
w1.9
44.89
48.15
57.13
54.32
65.55
69.90
the wall are smaller and there is less conversion to
the escaping wave. However, for a duct with curvature in the presence of inhomogeneity the guiding
properties would not be as good as for stronger
ducts.
An important result of Scarabucci and Smith
(1971) was that the leakage was much greater for
thin-walled ducts. This is confirmed by calculations
made for a and b equal to 49 and 51 km, respectively,
with the results shown in Table 5. Comparison with
Table 1 shows that, for 112= 1, the decay length of
the leading modes is of the order of a hundred times
less than for the thicker-walled duct.
Finally, in this section, we give results for a
trough (Table 6). a and b are again 40 and 60 km
respectively and the electron plasma frequency in
TABLE 6. Rxsu~rs FOR A TROUGH.
PROPAGATION
IS AT JUST ABOVE
68.69
the core is 120 kHz. But now we take the electron
plasma frequency in the outer region to be 120.6 kHz,
so that the density there is about 1% greater than in
the core, and we consider propagation at just above
the half-gyrofrequency,
with f = 13.8 kHz, corresponding to Y = l-91. As can be seen, the modes
are unattenuated.
7. CONCLUSION
The main result of this paper is that plasma density
enhancements can convey VLF wave energy almost
without loss, in agreement with observed behaviour.
However, we have not considered the effects of
curvature and inhomogeneity,
and there is much
scope for further work in this direction.
One
question we shall comment on briefly is that of
interaction between waves and energetic particles.
Interpreting q as the wave number, one may regard
the waves making up a guided mode as propagating
at an angle 3 to the uniform magnetic field, where
cos f? = k/q. For non-zero values of 6 one might
expect Landau damping to occur. Away from Y = 2
we have seen that modes cease when k = kpb. For
Y large, equation (16) gives that for the principal
wave in the core, kq is approximately equal to ksa2.
Hence the limiting value for 0 is given by:
cos 6 = kgb2/k,” = 1 - 6.
(311
1657
VLF modes in cylindrical duct
Thus for a 5 % enhancement the angle of propagation
of the principai wave in the core wouid be iess than
about 18’. Near Y = 2, the modes cease when, in
the core,
line. Hence if Y > 1 at the equator, as it must be
for the whistier mode to propagate there, then in
general it will greatly exceed one as the top of the
ionosphere is approached.
Thus the angle 0 for
possible trapped modes in a trough will approach 90’.
Acknowle&ement-This
work was carried out with the
aid of a grant from the Science Research Council.
REFERENCES
Allis, W. P., Buchsbaum, S. J. and Bers, A. (1963).
Waves in Anisotropic Plasmas, p. 134. M.I.T. Press,
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Angerami, J. J. (1970). Whistler duct properties deduced
from VLF observations
made with the OGO 3
satellite near the magnetic equator. J. geophys. Res.
75,6115.
B,.
Kennel (1966) studied low frequency (Y > 2)
whistlers, and showed that a positive growth rate
could exist for a significant cone of wave propagation
angles to the magnetic field direction, with maximum
growth occurring for propagation parallel to the
field. Values depend on the pitch angle anisotropy
and the hardness of the energy spectrum.
For
variation as the inverse square or cube of the energy,
he concluded that growth for angles of propagation
of at least 10’ is possible. Studies of higher frequency
whistlers by Brinca (1972) and Cuperman and Stemlieb (1974) show that as Y approaches 2 the growth
may in fact maximise for a non-zero angle of
propagation.
Thus a significant number of modes
may be involved at any one time. One may certainly
conclude that the attenuations we have computed
are likely to be negligible compared with amplification or damping via interaction
with energetic
particles.
These remarks have been made with especial
reference to propagation in density enhancements.
In the magnetosphere, the electron gyrofrequency
increases away from the equator along a given field
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