Planet. Spaec Sci. 1971. Vol. 19, pp. 1141 to 1167.
Pcrttamon Press. Printed in Northern Ireland
A THEORY
Department
OF VLF EMISSIONS
D. NUNN
of Physics, Imperial College, London, S.W.7.
(Received injinalform
1 January 1971)
Abstract-The
work attempts to give a theoretical explanation of the triggering of VLF
emissions by magnetospheric whistler morse pulses. Fist studied is the behaviour of resonant
particles in a whistler wave train in an inhomogeneous medium. It is found that second order
resonant particles become stably trapped in the wave. After l-2 trapping periods such
particles dominate the resonant particle distribution function, and produce. large currents that
are readily estimated.
The in phase component of current will produce growth rates about n times the linear value
when particles have been trapped for n trapping periods. A detailed analysis shows that the
reactive component of current is able to cause a steady change in wave frequency.
Next a realistic zero order distribution function is selected, and the growth rates and rates of
change of frequency are computed in a magnetospheric whistler pulse in the equatorial zone.
These results permit a plausible description of the triggering process. Explanations are offered
for a number of detailed features of the process. The theory seems to fit in well with observed
numerical observations.
1. INTRODUCTION
This work is a theoretical study of the phenomenon of morse triggered VLF emissions
in the magnetosphere. In recent years this has been extensively studied and documented
and the interested reader is referred to the literature for full details of the process (Helliwell,
1965; Smith and Angerami, 1968). Here we will only outline the main features.
High power VLF transmitters NPG at 18.6 kc/s and NAA at 14.7 kc/s transmit morse
code pulses, which propagate in the whistler mode around a magnetospheric field line and
are picked up at the conjugate point. Frequently, these pulses are observed to trigger
emissions. These are additional signals, amazingly narrow band in character, with an
amplitude usually several times greater than that of the incident morse pulse. These
emissions may last as long as a second, or even longer. Their frequency starts at the morse
frequency, or slightly above it by up to 400 Hz, and is observed to be steadily rising or
falling. Emissions may take a variety of forms, and upward and downward hooks are
often noted. The total frequency change during the course of an emission may be as much
as 10 kHz.
For a given amplitude of morse pulse a minimum time (-70 msec for NAA) seems to be
necessary for triggering to take place. Also noted is that triggering activity is strongly
favoured at frequencies close to half the equatorial gyrofrequency along the path of propagation.
At this point we should draw attention to various phenomena closely allied to artificially triggered emissions. These are triggering by natural whistlers and the innumerable
varieties of spontaneous and naturally occurring VLF emissions. Many factors may be
involved in these cases, and the present theory is confined to the more closed problem of
morse triggering. However the theory developed should be at least relevant to the broader
VLF emission problem.
2. PREVIOUS
THEORIES
There is general agreement among researchers that the cause of VLF emissions most
likely lies in the resonant interaction between energetic radiation belt particles and the
1141
1142
D. NUNN
wave, and that such interaction is most significant at the magnetospheric equator where
the resonance velocity is the lowest. It is of course possible that the observed emissions
originate in the ionosphere but more detailed satellite observations are expected to confirm
that this is in fact not the case.
An early theory was that of Dowden (1962) which was designed to explain spontaneous
VLF emissions. He supposed that they were caused by narrow bunches of high energy
particles oscillating up and down the field line. These could give rise to a linear cyclotron
instability and generate emissions by amplifying turbulence at locahsed positions and
frequencies. It is easy to see that such particle bunches could generate almost any emission
form. The idea is ingenious, and there is little doubt that such particle bunches are involved in the production of natural VLF emissions.
Das (1968) adopted Dowden’s idea that the emission cansists of selectively amplified
turbulence, and considered the case of morse triggering. EIe endeavoured to show that
the triggering pulse would in fact produce Dowden’s bunches. Using a loss cone distribution function, Das did in fact obtain much enhanced linear growth rates, but consideration will show that his concepts are really not valid for the case of a continuous narrow
band wavetrain.
A useful study was recently done by Abdalla (1970) who looked at the weak beam
exeitation of a whistler pulse in a homogeneous medium. She found that the growth rate
had the linear value at the front end of the pulse, but after one trapping period the power
transfer from particles to field changed sign, and died away with successive trapping
periods, Thus it seemed that triggering was impossible in a homogeneous medium.
An important aspect of the problem was recognised by Helliwell (1965) in his well
known phenomenological theory of VLF emissions. He pointed out that on account of
variation of both wave frequency and background magnetic f3eld, only a limited number of
the particles-termed
second order resonant- would be able to remain resonant with the
wave for an appreciable period. It was suggested that such enhanced resonance was
confined to a narrow zone called the generation region, and that depending upon the power
supply, this moved back and forth across the equator in such a way as to give rise to the
various emission forms. Although this theory is in agreement with Helliwell’s general
ideas, many of the details, in particular the computation of the resonant particle current,
are quite different.
3. BASIS OB’ THE THEORY
We assume throughout that the morse pulse and emission form a continuous narrow
band wavetrain propagating along the magnetic field line in such a way that the wave
vector is always parallel to the ambient field_ Such propagation is referred to as being
ducted and is believed to take place as a result of geld aligned density gradients. There is
reasonable evidence-mainly
from transit time data-that
many of the whistler signals
observed on the ground travel in such a mode (Helliwell). In this paper we shall not
consider the recently discovered unducted mode in which k is not parallel to B,, (Walter,
1969; Scarabucci et al., 1969).
The problem at hand divides itself into two parts. As the wavetrain propagates along
the field line it interacts with resonant particles giving rise to currents. These have to be
computed by integrating over the resonant particle distribution function in the normal
way. Secondly, we must show how such currents produce an emission and elucidate how
they cause the wave frequency to change.
A THEORY OF VLF EMISSIONS
1143
The problem is rendered very complex because of the inhomogeneity of the medium.
A resonant particle travelling through the pulse will see wave frequency, wave number k
and background magnetic field all steadily changing with time. Only particles which come
close to satisfying a second order resonance condition
h---n
co-n
do,
dv,,,
-=---.
-=
k;
(1)
v, = v,ea
K
dt
dt
KZ
will be able to stay in resonance with the wave for an appreciable time.
The electrostatic analogy
Some valuable insight can be obtained by studying the much simpler but closely analogous case of the weak beam excitation of an electrostatic pulse in which wave number
is a linear function of position (Nunn 1970a, b). It was found that second order resonant
particles became stably trapped in the wave and could stay in resonance for very long
periods. After one or two trappingperiod~ such particles were found to dominate the distribution function about resonance and to make a very large contribution to the resonant
particle current. Such currents are proportional to the time for which such particles have
been trapped, and it was found that a pulse which kept particles trapped for about 10
trapping periods, say, would exhibit a growth rate about 10 times the linear Landau
value. In addition, a reactive component of current was found which was exactly what
was needed to cause the wave frequency to undergo a steady change.
These ingredients of very large growth rates and a steadily changing wave frequency
are exactly what we require to explain triggered emissions. Accordingly, the theory to be
presented is based on the assumption that the whistler amplitude is sufficiently great
such that second order resonant particles may be trapped in the equatorial zone for several
trapping periods.
4. RESONANT
PARTICLE BEHAWOUR IN AN INHOMOGENEOUS
MAGNETIC FIELD
Our first task is clearly to examine the behaviour of resonant particles in an inhomogeneous magnetic field. Noting that the second order resonance condition Equation (1)
involves the gradients of Be and o, the simplest appro~mation to the whistler emission
field that will best illustrate resonant particle behaviour is one in which B,, o and k are
taken to be linear functions of position. This is of course much simpler than taking the
full parabolic form for the field, and we shall see that it is sufficiently general for our
purposes.
We require a constant amplitude whistler with k parallel to B,, and with linearly varying
B, and o. For small values of the gradients i.e.
&+z
Q ko
aB,/az 4 kB,
we may write the fields as;
E@=
E, =
E, =
B, =
Ecos#
Esin$i
0
- Ek(z, t)c sin #u(z,
4 = Eke cos #CO
B, = B&(z)
(2)
t)
D. NUNN
1144
where z is in the Be direction, The quantity 4 is the phase of the wave. For &.#z # 0
it is c&arly going to be diffcuit to give a satisfactory expression for 4 as a general function
uf z and t. Rowever if we regard 4 as being just the phase as seen by a particuhtr particle,
we do know that in some average sense the following will be true.
d+
rr--
-k(E,
tjv, + w(z, J).
In reality dads wih have an irregular v~~~t~on about this value pro~rt~~~~ to the frequency band width of the emission at that point. Howcvcr in what fohows we shall be
concerned with phase stable particles, and small irregularities in phase are not expected
to be significant. Thus we may represent the phase as seen by a particular particle as
being the foIlowing trajectory integral
For simplicity we shall assume that the cold plasma density is proportional
which gives the following dispersion relation
83 = ~~~~~~~ - CS)_
to B,,
(4)
The group velocity simply becomes:
V, = 2/A(l - w/Q)%P!
(5)
Provided that the eh&r~~~ Larmor radius is much fess than the scale length of & the
equations of motion of the particle can be shown to be, using the usual symbols
It is ~~~~~~i~~t to d~~rne~sio~~~~ these equations using~e values of~,~~~~~p~r~n~~g
to the point z = 0, which is where the trajectory integrations witi start. Assuming that at
this point w = 6, k = k, aad @= 1, we define dimensionless variables as folXows;
1145
A THEORY OF VLF EMISSIONS
Substituting from Equations (2) and (7) into Equation (6) we obtain dropping primes:
Defining y = V, + iv, this reduces to
-dy = - R(1 - kVJw)&
dt
+ k y V, g/j?
4 i&/a
(9)
We suppose a spatial variation of frequency of the form
cO(z, t = 0) = 1 - &z.
The entire emission field is advanced along the field line at the group velocity Z(1 - a).
Thus:
~(2, t) = 1 - [(z - 2(1 - a)t).
(10)
Note that in the generation region of an emission there will also be an absolute rate of
change of frequency dwjdt. As far as resonant particle behaviour is concerned, the system
is exactly as above if we define as effective & as follows.
(11)
The dispersion relation now becomes
kZ = ~(1 - a)/(1 - oat@).
We also specify the z-dependence of magnetic field,
(12)
B=lJrgz
and the rate of change of wave phase is given by
(13)
d$fdt = CO- kV,.
(14)
Equations (9-14) are those requiring to be integrated to determine resonant particle trajectories. Before discussing their numerical integration we shall investigate analytically
some of the properties of second order resonant particles.
The condition for exact second order resonance
We now enquire which particles are able to satisfy instantaneously
order resonance conditions :
v* = Y,l?a
dV,/dt = dV,,,/dt,
the exact second
1146
D. NWNN
This is easily found as follows. Using the same dimensionless units as before, the resonance
velocity is given by
Kee. = (a - ~~~~~~
and its rate of change as seen by a particle can be shown to be
The equation of motion, Equation (Q furnishes the rate of change of V,.
NC/w Irl cos P - jy[* q/2/3
V; =where :
P = tan-r( V,/ V,) - 4
is the phase difference between the perpendicular velocity y and the ekctrk field. Specialising to the point z = 0, the second order resonance conditions thus reduce to
cos P = 1-C tana IK,~ - Al/tan ie,I
(13
where a9 is the pitch angle. Thus at a given point, exact second order resonance is found
for particles which lie on a given line in velocity space. This line is in the V, = V,, plane,
and its form depends on the three quantities qfR, E/R, a. Note that if the coeiTicients
A and C are of the same sign and also v%! > O-5, second order resonance is impossible
and the curve disappears.
Phase stabilityand particle trapping
As in the electrostatic case it is not enough that a particle be initially second order
resonant. We must enquire in detail of its subsequent motion. We are particuIarly interested in whether particles are phase stable or not -in other words, if they are displaced from
the resonance line, do they oscillate about it or do they fall right out of resonance? Of
course, the exact behaviour of resonant particles will be very complicated, but it .is not
difficult to extract the dominant terms that impart overall stability or instability.
We proceed as follows. The equations of motion may be written:
dfjdt = kjV - V,,,) + g sin P(1 - kV&4
dlrl2=-_R(l
dt
1
- kV,jw) cos P + z
(W
IrlV, waz
B
We assume throughout this paper that R 4 1, or that the resonant width is very narrow.
(16 Q 1). Let us now consider a particle whose z component of velocity is roughly
A THEORY
OF VLF EMISSIONS
1147
within the limits I’,, & fi.
We deal with a quantity P,,(t), which is the phase of the
second order resonance line at the particle’s perpendicular velocity 1~1,and at the particle’s
instantaneous position. Note that PO can be imaginary if the resonance line does not
exist at that particular value of 171. Neglecting some small terms in Rli2, we may write:
ag28.
cos PO - I# az
I
Defining V’ = V, - V,, we then get
dV’
-=
dt
Rk
- ;
171 [cos P - cos P,].
Since for trapped particles V’ N l/i,
approximation, as
Equation (18) may be written, to the same order of
dP/dt N kV’.
On eliminating V’, and dropping a small term in dk/dt, we obtain the following differential
equation for P
d2P
dta+
Rk2
co Iyl(cos P - cos P,} = 0.
(21)
A first approximation to the detailed motion may be obtained by assuming that amplitude R, k, co, Iyl, and POare all slowly varying functions of time, as compared to a trapping
period R1j2. In fact these quantities will tend to have a time scale of l/R. So, neglecting
the time dependence of these quantities, and confining our attention to fairly small amplitude oscillations about the resonance line, there results:
d2P’/dt2 - *
0
(sin P&P = 0
where
P’ = P - PO.
For 0 < P,, -c w the solution will be exponential in character, and particles lying initially
on the right hand side of the resonance line, will fall away from it and become non resonant.
On the other hand, for 0 < PO < - T the solution is oscillatory in character, and second
order resonant particles oscillate about the line and are thus stubZy trapped. The nature of
particle motion within the trap is illustrated in Fig 2. Particles describe elliptical trajectories
in the coordinates P’, V’ around the resonance line. The frequency of these oscillations is
seen to be
cotrap-
R1j2k
IF?
or the usual ‘trapping frequency’ of the homogeneous problem.
tion in V’ is given by
The amplitude of oscilla-
1148
D. NUNN
fk3.
Fro. 2.
1.
A
SKETCH
TRAIECTORIES
THE
OP THE
OF
METHOD
WAVE
STABLY
PACKET
TRAPPED
OF CONSTRUCITNG
USED
SECOND
THE
FOR
THE
ORDER
TRAP
M
TRAJECTORY
RESONANT
VELOCITY
INTEORAlTONS.
PARTICLES,
SHOwtG
SPACE.
which is -Rl’a as expected. For large amplitude o~illations the motion becomes anharmonic, and the trajectory of the particle will be roughly fixed by the relationship
dV’
dp’=
- y
(cos P - cos PO).
It is not difficult to see that if the particle trajectory goes beyond the unstable branch of
the resonance line P =- PO it will escape from the trap and fall out of resonance. It is
now possible to construct the region in velocity space occupied by trapped second order
resonant particles at any given position. One takes the instantaneous resonance line, and
for each value of 1~1constructs (from Equation (23)) the limiting trajectory that just reaches
P =- Pw This is illustrated in Fig. 2. The resultant surface is the outer boundary of
the trap.
Note that superimposed upon these oscillations will be steady rate of change of 171due
to the resonance with the wave and also due to the magnetic field gradient. Some computed
examples of the time variation of P and ~121
are shown in Fig. 3. Note also that the resonance line itself will in general be a function of time. However, provided it moves slowly
with respect to a trapping period Rlf2, particles will be able to continuously circulate
around it ~thout ~~e~ty~
The numerical integration
By way of confirmation of the foregoing ideas we now turn to exact numerical integrations in an actual wave train-in fact using the fields already outlined. We take a fairly
A THEORY
1149
OF VLF EMISSIONS
+-270.
#I-SO’
+=1830*
FIG. 3.
COMPUTED
EXA~LESOF~ETI~
VA~ATIONOFP~CH
ANGLE ctg,
AND
TRAPPED RESONANT PARTICLES.
The curxs are two separate examples of the resonance. line.
PHASE P PER
large amplitude R = 0*0004 and assume that the pulse extends from z = 0 to z = 2000
(see Fig. 1). This means that a resonant particle will spend about 10 trapping periods in
the pulse.
Our task is to numerically integrate Equations (9-13). The computational technique
is the standard one for such problems and will not be discussed here. The integrations
start at z = 0 and follow the trajectories backwards in a stepwise fashion until z = 2000
is reached. A network of particles is selected in the y plane, and for 5 values of V, equispaced across the resonant width. All this is done for a wide variety of the coefficients
u, q, 5. Noting that the rate of change of particle energy goes as:
rvocIV11COSP
a suitable indicator of the trapping history of a particle would obviously be
where A W is the total change in energy of a particle as a result of passing through the
pulse. The quantity A W* is in the nature of a trajectory integral of cos P, and the numerical
output of the program takes the form of a plot of this quantity over velocity space in the
neighbourhood of resonance.
Results of the trajectory integrals
The results are readily understood in the light of our analytic study of resonant particle
behaviour. A typical plot of the energy change A W* in the resonance plane is shown in
1151
A THEORY OF VLF EMZSSIONS
270e
I
I
60
40
/
20
0
.__
........... “,+JR,Z
-
Resonance
line
AW*).
graph showsthe boundary of the region in the Vl plane for 3 valuesof V,. q/R = -@6,
b/R = @25,a* = 0.5.
FIG. 5. REGION IN VELQCXTY SPACE OF EPFETIVE PARTICLE TRAPPING (LARGE
The
----
-. ....
I
FIG.
6.
TRAPPING REGION FOR
q/R =
Using the same system of dedimensionalisation
B = B(z)
v .__
r..-JR/2
vr_+
,/R/2
Resonance
line
2, E/R = -1.275, a* = 0.5.
as before we have
R = R(z, t)
Co = o(z,
t).
We now enquire as to the distribution function about resonance, F,,, at some arbitrary
point, z = 0 say.
We consider a time independent zero order distribution function F. that is a function
D. NUNN
1152
0’
----
V,-JR/2
o-5.
120’
v,-
JR.e
Res3nafice
----
RG.
8.
TRAPPXNG
REGION
of energy and magnetic moment only.
POR
?j/ff
=
-0.2,
E/R
line
2=
0.1, a* = 0.5.
A THEORY
1153
OF VLF JZMISSlONS
180’
----VW-JR/2
..._......
,,.+
-
FIG. 9. TRAPPING
REGION FOR
E/R= -0.45,
JR,2
Resonance
line
T/R = 06,a*
= O-5.
0.
90’
180"
---- I’,,,-JR/P
........... V,.,+JR,2
Fzo. 10. TRAPPING
REGION FOR
Resonance
line
q/R = 2,5/R= -1.5,
a* = O-5.
where 6W, 6,~ are the total changes in energy and magnetic moment undergone by a
particle up to that point as a result of interacting with the wave. Now if particles are
trapped for times of order l/R 6,u, 6 W will be large, but this case normally represents a
very powerful wave. In the VLF emission problem we are justified in neglecting (SP)~
(~3~) and higher order terms. From the equations of motion (Equation (9)) we quickly
derive
I@ = --RcosP Jyl
&f2
= --R(l
9
- kV,/o) cos P. lyl/p
(27)
D. NUNN
1154
whence
8w=
f
RcosPIrldt
R(1- kV,/o)
6,~ = -
cos P//3 IyI dt.
s
(28)
Assuming that the wave amplitude is sufficient such that particles are trapped for at least
several trapping periods, the results of the previous computation strongly suggest that at
any point I;,,, will be dominated by a core of well trapped second order resonant particles
centred about the instantaneous resonance line. This being so we may approximate
Fr,, as follows. For particles outside the trap 6~~ 6 Ware small and we may put F,, = F,,.
For particles inside the trap we need only integrate over the time for which particles have
been trapped replacing cos P by cos PO and V, by V,,. Hence
We may write this as
Fwa = Fo(W, p) 4 R ~0s !‘o
(29)
which del!nes an effective trapping time 7,&l, P, V,). Except in simple cases T,!~ may be
difficult to estimate accurately, without resorting to trajectory integrations. However, on
account of the continual stirring of particles within the trap, we do except 7,ff to be fairly
symmetrical about the resonance line, and to fall off towards the outer boundary of the
trap.
6. THE RESONANT
PARTICLE
CURRENTS
Particles with their parallel velocity close to the resonance velocity V,, can give rise to
transverse currents whose frequency and wavelength are very close to that of the wave at
that point. Such currents will be resonant with the ambient plasma, and can thus cause
wave growth.
The current due to trapped second order resonant particles is well estimated by integrating expression (29) over the volume of the trap in velocity space. There will also be a
sizeable contribution to the current from non resonant particles, but owing to dephasing
effects, this will be parallel to Bo. Thus we may write, since F. is phase independent
9=jR+ijr=
-e
N
We now confine the integration of F,,-F.
FrC0’ The power transfer to non-trapped
r’lrl(4es - Fo)d Irl dv, dp.
(30)
to the particle trap. This merely approximates
particles is important and is not ignored in
making this approximation.
The current has been divided into two parts, the in-phase component J,, and the reactive
component Ji. Clearly 3 will depend upon F. and the resonance line in a rather complicated
way, and we shall consider this later. At this point we will discuss one or two rather
important general properties of 9.
1155
A THEORY OF VLF EMISSIONS
First there is a general proportionality to trapping time T,!!. Other things being equal,
both J, and Ji will tend to increase steadily downstream from the front of the wave packet
(in the direction of I’,,). Superimposed upon this linear spatial dependence will of course
be a variation due to the changing trap geometry along the pulse and to other complex
effects.
Another significant feature is as follows. The current g is caused by a jet of phase
stable particles. The phase difference between 9 and E will be closely determined by the
local resonance line which changes only slowly over a trapping length. Thus provided the
wave field remains ‘slowly varying,’ this phase difference will remain nearly constant, and
be maintained under fairly general conditions.
Perhaps a more important point is that .? is produced by a small number of particles,
closely confined in velocity space, that have interacted with the wave over an appreciable
distance. The wavelength of the current produced cannot change quickly merely because
of the inertia of the particles producing it. One thus expects 1 to be a very narrow band
in character, even when the ambient plasma has irregularities in it.
7. TIME DEVFLOPMENT
OF THE WAVE FIELD--GENERAL
THEORY
We now consider in general terms how resonant particle currents of the type we have
described can modify the wave field. Being resonant with the ambient plasma the current 3 will radiate like an aerial and generate additional fields which add on to the existing
ones in a continuous fashion-the current g of course being all the time determined by the
instantaneous total field in the manner discussed. Provided the wave parameters vary very
slowly over a wavelength the field radiated by g will be proportional to it in magnitude
and with the same phase.
Let us consider the field addition process taking place in a limited section of wavetrain.
In accordance with the previous discussion we suppose that 9 increases linearly with z and
maintains a constant phase with respect to E. Take an initial field of the form
E = E, + iE,, = E(z)e-“@.
After a small element of time we add on the current generated field to give:
J%OT
=
e-i)NfE<z> +
@4jd4
-(E(z)
+
i(~A)j,(d)
+ (sA)jR(z))e-*@e”’
The local rate of growth of field amplitude is proportional to the in phase component of
current J, as expected. The combined field is seen to have a small wave length shift proportional to i?/az(JIIE). (This is illustrated in Fig. 11.) Subsequent resonant particles will
respond to this new wave length and shift it still further and so on. Thus we expect that the
reactive component of current is able to cause a steady change in wave number and wave
frequency.
Analytic model
The foregoing argument is helpful in providing a physical picture of the way in which
the frequency change occurs, but it is clearly .far from rigorous. It is however possible to
deal with the simple case of a constant amplitude wave analytically. We proceed as follows.
1156
D. NUNN
Fro. 11.
THB
FIELDS,
LABELLED
J*, TO
E WITH A SHIFTED WAVELENGTH.
The wave length shift can be seen to be. proportional to a/az(.&/E).
SKETCH
EXISTING
SHOWING
FIELD,
THE
LABBLLED
ADDITIGN
E,,
TO
OF
GlVB
CURRENT
A NET
GENERATED
FIELD
We choose to look at a limited section of wavetrain-of
the order of a trapping length
say. One assumes that although the inhomogeneities are important in determining the
resonant particle currents, for the purposes of analysing the effects of the current upon the
wave field one may assume the medium to be locally homogeneous. The reason for this
is that the cold plasma particles are stationary and do not sample the field gradients in the
same way as the resonant particles do. In the following analysis terms in 6’k/az etc are
truly of order R and thus unimportant.
We thus take a cold background plasma with constant gyrofrequency Q, and plasma
frequency II. The electric field is represented in terms of an amplitude and a phase factor
thus :
,!? = En + iE,, = E(z, t)eti(@)
(31)
and
A single differential equation for the time development of Scan be obtained from Maxwell’s
equations and the equations of motion of the cold plasma particles
(32)
The reactive and in-phase components of current are defined by
(33)
In accordance with our previous discussion we expect J, and Ji to increase linearly with z,
but to be otherwise only weakly dependent upon z and t. This holds on& provided E
remains narrow band and satisfies the ‘slowly varying’ criteria. We now substitute for
&, and J!?in Equation (32) and neglecting small terms in l/c2, there results two differential
l
This analysis is conveniently
done in dimensioned
variables.
1157
k THEORY QF YLF EMISSIONS
ak
--2k(u-Q)pj
Izki3
at
-I-
ilk al??
----
a2 at
$ --s2)E -
g*
=
ak a&-_ 2k PE
23ilG
-zz
azat
$ w(fiJ -
4x
--”
-fJJ
t
(u-q,.
Q)jR
(34)
(35)
Here u and k azu:simply defined as being
uf = ~~~~r
k =-
~~~~~*
Terms in ji andji have been neglected in comparison with ccjn$ @jr, l[n seeking solutions to
these equations it was found highly dangerous to eliminate higher derivatives, and also
that the initial time behaviour of a particular field confIguration could be misleading
indeed. The only ~~~b~~~ty is to find exact foag time solutim5 of a f&r@ simple ohara&er_
We assume that E, k, & are ah independent of z, and that JS varies only linearly with z.
In this case Equations (34) and (35) reduce to
8!$ @a -i- TP/c”) + 2kE alk/at = $ m(oJ - !2)ja
~~~~(~ - 0) - ~~~~~~ = - ;
ttf(w - Q)jp
(36)
(337)
We now assume that the resonant particle beam is weak. If e is the smallness parameter
associated with this, then we see that the quantities I?, k, Jr and Ji are all of first order in E.
Let us define a proper frequency from the dispersion relation
k~{Z - $5) = &Se/C9
(3%
and put
cc)= CZ“+ 01.
Noting that 09 is also of order is*we may substitute into Equations (36) and (37) and ehminate terms in 89 From Equation (37) there results
47T S(& - f$)jz
(0~ = ca &@a + rP/c2) *
(39)
Differentiating with respect to z there results
2k
ho1
-...“S~--~=ar
az
47r cz(G - G)Z a
rz jp
IIs
GE
0
M&e that the z-independence of k is preserved, as is essential for a Iong time sofution.
The rate of change of frequency is obtained at once from its defining equation
&G 87&(&“J - $)a a
.---V=
az jz*
at
&z%EP
(41)
1158
D. NUNN
Equation (36) becomes
a_E
ar=
4?T&J(c;i
- Q&n
rP+ky@
~E~k~~t
(42)
-v+riyca
where again we note that .&is not a function of z. Some consideration will reveal that
iTkJ&jk 4 aEji?tJE
which reduces the growth rate to
We have thus found a long time solution as follows. A wave with E and k independent of
position in a locally homogeneous medium is excited by a constant in phase current and a
linearly z-dependent reactive current. Quite rigorously, to first order in E, the wave undergoes a steady change in amplitude, wavelength and frequency as given by the above expressions,
Such an exact solution certainly lends support to our general argument for a shifting
frequency, and suggests that a more.generaX expression for 6 might be
Note that such long time expressions for 5 will in fact only hold for simple systems of the
kind we have just studied where a/&(J,lls) is itself independent of position or only very
slowly varying.
In more realistic cases of finite length pulses, the inertia of the trapped particle beam
will become irn~~nt
and will tend to act in such a way as to produce an averaged &
common to the pulse as a whole. This question is far too complicated to discuss here.
8, COMPUTATION OF GROWTH RATES AND RATES OF CHANGE OF
FREQUENCP IN A MAGNE!I’OSPHEBIC !VHBl’LRR PULSE
We now consider a whistler pulse promoting
along a ma~etospheri~
Geld line.
Assuming that the wave parameters are slowly varying with respect to a trapping period,
we endeavour to compute the currents due to stably trapped second order resonant particles,
using a realistic zero order distribution function F&V, ,u). Next we compute the growth
rates and rates of change of frequency which result from these currents and show how they
depend upon wave frequency, position along the field line, the rate of change of frequency
of the pulse, wave amplitude, and F,. These results should give us a considerable insight
into the triggered emission problem.
Units
In what follows we require suitable units for velocity, time etc., so we now dedimensionalise with respect to o,, ane half the equatorial gyrofrequency, and ko, the corresponding wave number. The variation of p along the field line is modelfed in the usual way
/3 = 1 + ; xz*a;
where x is taken to be 2 . lo-*.
z* = k$
(45)
A THEORY OF VLF FMISSIONS
1159
In presenting these results we need independent variables related to invariant properties
of a wave pulse travelling along a ma~etospheric field line. Thus frequency is referred to
equatorial gyrofrequency as follows;
cc* = o/B,*.
The flux of energy down a duct would be proportional
to
Eak/oj3.
Accordingly we construct a dimensionless amplitude factor R* depending only on power
flux;
R*=e-
E
kko
M J Z&CQ’
The frequency gradient is referred to at,, k. as follows
[* = - ih/&/kooo.
We numerically compute the growth rate as a function of R*, &*, ct*, z*, using Equation
(43) and substituting for .T, with Equation (30). It is apparent that some kind of simple
appro~ma~on for retL is going to be required. We make a rather arbitrary assumption
about the average extent of particle loss from the trap and suppose that particles whose
trajectories take them beyond P = --PO!2 will not be effectively trapped. Thus at any
point we may define a ‘reduced trapping volume’ in velocity space (see Fig, 13). Inside
this volume we assume that ~~~~is constant with a value R*-l/*/l V.,j. We are thus computing a growth rate per trapping length and the actual growth rate will be obtained by
multiplying by ~~~R~-~i~, where AZ* is the average distance over which particles have
heen trapped. The growth rate is best expressed as a multiple of a quantity which estimates
the linear cyclotron growth rate
‘* --
1T200”
koiVo
[~VJhF6]~z_vre~
Thus we plot a quantity:
(49)
The rate of change of frequency in the pulse is a much more complex problem than the
growth rate. Here we compute a much simplified expression ycu*. This is the local rate of
change of frequency that would take place if the variation in E and in the resonance line
itself were unimportant.
In other words, we take Equation (41), substitute for Jt with
Equation (30) and assume that the z-dependence of Ji comes solely from that of T,~, i.e.
a/az Teff = -~/IK,I
and
ym* = d~ld~t~o~~
(501
To compute y*, yu* we require a reasonably realistic zero order distribution function
Fo. The one used is shown in Fig. 12, as a function of equatorial VL and V,. It embodies
a rough fall off with energy as E- a*5,together with a pitch angle dependence of the form
log (sin a/sin %), where or, is the pitch angle.
D. NUNN
1160
Distribution function Fo
fiG. 12. THIZ ZERO
ORDER
DISTRJBUTION
FUNCTION
AND
F,,
AS
A FUNCTION
P*O
13. EXAMPLE
OF THE REDUCED
SECTION
TRAP USED TO COMPUTE
IN
THE
v,
Reduced trap;\/, -V,.,
--Ro.
OF EQUATORIAL
v,.
RESONANCE
Trap boundary
Particle
trajectories
THE GROWTH
RATE y* (CROSS-
PLANE).
Results
The results for y*, ym* are illustrated in Figs. 14-16, and on the whole these graphs will
speak for themselves. A number of important comments however should be made. Referring to Equation (30) it can be shown that the instability comes from a positive value for
aF,&,
whereas the usual fall off with energy aFo/a W will tend to give wave damping. Thus
the phenomenon of triggering instability is due entirely to the presence ofa loss cone, and the
A THEORY OF VLF EMISSIONS
1161
fxc-wool
4
3
2
I
0
-I
-2
-3
-4
-4GcQ
4om
FIG. 14. PLOT OF y+ AS A FUNCTION OF Z*,E*,
FIG. 1% PIsoT OF pm* (Xlm)
ASA
FOR cc+= O-5; R* =
FUNCTION OF z*, (', FOR a* =@5;
moo
0$~0002.
R* ==0-~2.
sharper the pitch angie gradient the more unstable the system will be. This result is in
sharp contrast to Helliwell’s theory. Even so though, if the fall off with energy W&?W is
too rapid, the instabi~t~ may be quenched.
A typicai graph of the dependence of y* upon z*, E*, is shown in Fig. 14, for R* =
090002, a* = O-5. This kind of plot was found for all a*, and for most reasonable choices
of F,,. Variation of R* leaves the contour pattern almost intact, but the extent of the pattern
along the z*, 5* axes is directly proportional to R*. The frequency dependence of the
1162
a.
NWNN
magnitude of y* is shown in Fig. 16, and this is very critically dependent upon F,,. Generally though y* falls off at low frequencies because of the diminished supply of resonant
particles, and also at high frequencies because of the thermal character of the distribution
function at low energies.
Note that the maximum growth rate is found close to the equator and for rising or
falling wave frequencies. This result is also diRerent from IIelliwell% predictions.
The graphs show that y* is typically of order unity, In other words, second order
resonant growth rates are of the order of the linear growth rate when 78ffis only one trapping
period. The procurement of very large growth rates from second order resonant particle
trapping is thus entirely dependent upon keeping particles trapped for many trapping
periods.
The results for yQ1*are very similar in their dependence upon z*, P, F*, a*, except
for an overall dependence upon R*lia. A sample plot of yw* is shown in Fig. IS for R* -=
O-00002, a* = OS. Note the two symmetrical zones in which the wave frequency is made
to rise or fall, and that the rising zone is on the receiver side of the equator.
At this point we note that the entire description of VLF emissions being developed
remains the same if the direction of the static magnetic field is reversed, This merely
changes the direction of rotation from right to left.
9. THE TRIGGERING PHASE
The in&l growth rate
We now turn our attention to the triggering phase of the process, and investigate the
growth rates to which the morse pulse is first subjected on entering the equatorial zone,
During this phase the pulse frequency remains close to that of the incident morse, and we
may put R* = R* in, a* = 04, E* = 0. Inspection of the r* graph (Fig. 14), points to
two main regions of growth, one on each side of the equator. The actual local growth
rate will be proportional to 7Bff, and will thus reach a maximum on the transmitter side of
each zone. This maximum rate of growth is estimated by
where AZ* is the maximum effective trapping length, which will be the characteristic
width of the growth region, and &,X is the maximum value of y* along the z* axis. We
1163
A THEORY OF VLF EMISSIONS
have already noted that AZ* is proportional to R*, and so the maximum initial growth
particles
rate will vary as R *1’6. Now when R* is small enough (R* = R,* N O,~l),
only stay trapped for about one trapping period. This represents the lower limit of validity
of the present theory. The growth rate is then conveniently written as
dR*ldt -
yoy;8x(R*/&*)S’2.
(52)
Clearly Equation (52) prophesies an almost explosive kind of instability. However, as
the morse pulse passes the equator, amplification occurs, but the enlarged signal is continuously advected away. The equation governing the time development of R* in the triggering zone would be roughly of the form
&R*
w Y&&(R*/R,*)~‘~
-
V*(R* - RIJRO*
5OoR*
*
Once R* reaches a certain critical value the growth rate is sufficient to form a self sustaining
generation region. The growth on the receiver side will develop into a riser, with a failer
on the other side. A given morse pulse may excite one or both of these, and once the frequencies have separated they will be separate entities, uninfluenced by each other or by
subsequent morse.
An interesting detail is that any element of morse wavetrain must pass through both
growth regions, and thus the wave amplitude must always be greater in the ‘rising’ zone.
This explains the noted tendency to trigger risers rather than fallers.
The triggering delay
The building up of the amplitude R* to a self-sustaining value takes a finite time, known
as the delay time. The latter is difficult to estimate owing to the complexity of the process.
We expect it to be mainly a function of morse amplitude R$,*, and also ye?&.
Clearly, the greater Rti* is, the shorter the delay time will be. An interesting point
however, is that the delay time does not become indefinitely small for very large R,*. This
is because between the first appearance of the wave field and the establishment of the
maximum growth rate of Equation (52) there must be a time lag, which will be of the order
of the time for which resonant particles may be trapped at that amplitude (T&.
The delay time will also depend critically upon the product yOyzax. However, time
delays at constant R* seem to be surprisingly consistent in their behaviour. The answer to
this may be as follows. The linear growth rate for broad band turbulence will vary with
distribution function in much the same way as ye?&.
Kennel and Petschek (1966) have
shown that there is a certain upper limit to the linear growth rate above which the system
is absolutely unstable. It is thus possible that the observed minimum in delay time corresponds to the maximum stable value of yOygsx.
The frequency dependence of triggering
Observations of the frequency dependence of morse triggering show that this is quite
sharply confined to values of cc* just below 0.5. The present theory does not single out
any particular frequency, and thus an explanation for this must be sought either in some
propagation effect or in some singular property of FO. Two possible explanations are;
1. The frequency a* = 05 is a limiting one for ducting, (He&well, 1967) and above
this value a pulse will become unducted in the equatorial zone causing considerable loss of
1164
D. NUNN
wave energy. Since y* tends to be a rapidly increasing function of a* in this region, one
would then expect to see triggering confined to frequencies just below a* = 0.5.
2 Electrostatic diffusion by whistler turbulence is particularly effective at V, = + 1.
One might expect to find a sharp fall off in F,, above this value of V,, and this would give
a pronounced peak in y* at a * = 0.5. The peak in graph 16 was obtained by making the
gradients of IJ,, discontinuous at V, = + 1.
The o&vetfrequency
One feature of emissions often observed is the positive offset frequency for both risers
and fallers. This is easily explained as follows. Let us consider the resonant particle
current that first appears in the morse pulse in the equatorial zone. This is sketched in
Fig. 17 for the case R* = 04IOOOO5,a* = 0.5. At z* = 1000 the current is in antiphase
to the electric field, but steadily rotates to become almost entirely reactive at the equator.
Initial currents in t&e
E
Z*----~
EQ
+
f
J rei
t
,
t
I
t
u
1
V
,_ m
18,
I
Z*=-200R%?;
FIG. 17. SKETCH OF THE AMPLITUDE AND PHASE OF THE SECOND ORDER RESONANT PARTICLE
CURRJ3Nl-STHAT PIRST APPEAR IN A MORSE PULSE IN THE EQUATORIAL. ZONE.
These currents have on average a shifted wavelength-thus
giving rise to frequency offsets.
The direction of rotation is the same on the other side, and at z* = -1000 the current
is again in antiphase. On both sides of the equator this current will have, on average, a
fractionally shifted wavelength of
ak 742
-,-,+k
AZ’
&, Ro*I%*.
This shift is inversely proportional to I& *. The first extra fields generated by these currents
will have this shifted wavelength. For R* = 04000025 say, dk/k - O-003, and the corresponding frequency shift will be &.I N 100 Hz, which is the right order of magnitude for
the offset frequency. This seems to provide a fairly simple picture of how the offset frequency arises, but the subsequent frequency development of the system is rather awkward.
The combined field of the morse and that generated by the currents will have a beating
amplitude, and the resonant particle current can then only be found by doing the requisite
trajectory integrals.
One point to be noted is that the wave train frequency cannot separate? from that of
the morse until the power input is sufficiently great to form a self sustaining generation
t Two frequencies are deemed to have separated when particles resonant with one of the waves are
negligibly tiected by the presence of the other.
A THEORY
1165
OF VLF EMISSIONS
region. Until that stage is reached the phase angle between E and 1 will adjust itself automatically in such a way as to hold the wave frequency constant.
10. THE
GENERATION
REGION
After triggering has taken place, the wave train settles down to a semi equilibrium
condition characteristic of the mature emission phase. This is the kind of situation described by Helliwell, and is illustrated for a riser and a faller in Fig. 18. The head of the
wave train is known as the generation region and is where resonant particle interaction is
most energetic and effective. The region is character&d by a very large power input,
which must be large enough to maintain the wave envelope in a constant position. Additionally, the frequency in the generation region is constantly changing. Downstream the wave
particle interaction is weak, and the generated waves merely stream down the field line to
be picked up by the receiver as a rising or falling frequency.
o@j/pF~,
7*,
EO
-1000
2000
1000
(a) Generation region of riser Q*=0.5
4*= OT)OOOO5
(b) Generation region of failer a*=O.5 [‘=-0~000005
FIG.
18. ANTICIPATED
PORMS
OF THE GENERATING
REGIONS OF A RISER AND
FALLER.
The g.r. is sustained by a jet of stably trapped second order resonant particles. However,
owing to the fairly rapid change of field amplitude with distance true ‘trapping equilibrium’
does not obtain, and strictly speaking the resonant particle currents can only be obtained
by doing the appropriate trajectory integrals.
However, the y*, yW* diagrams can still be of great help. Firstly, in order that the
power supply be sufficient, we expect the values of z* and E& for a g.r. to roughly correspond to one of the well defined maxima in y*. This suggests that the g.r. for both riser
and faller will be located very near the equator. The actual growth rate at a point in the
g.r. may be very roughly estimated as follows. Assuming trapping equilibrium, we have;
y - y,,y*([*, z*, R+)( rR*dz*) R*-l12
\J
I
where the integral is from the given position to the receiver end of the g.r., and y* is the
local value. In fact y will always tend to be somewhat larger than this value, because
the current will tend to ‘overshoot’ the field.
The estimation of the rate of change of frequency of the g.r. is rather more difficult.
Owing to the inertia of the trapped particle beam, the phase difference between E and g
1166
D. NlMN
will necessarily adjust itself in such a way as to give a common rate of change of frequency
for the whole g.r. We expect &&to be obtained by some kind of average of yo* over the
g.r. (Probably an average weighted according to rate of power input.) Thus
5:~ = YOY~*(& R*, z*>/wo(lv,l + I v,,l).
W)
Note that since ya* is a function of t&, &$ is a function of itself. Note that the above
relationship is exactly what imparts seIf consistency to the wave particle interaction. The
wave field with a changing frequency organizes the particles in a particular way, and generates the very currents which are required to make the frequency change at that rate.
Inspection of the ya* graph shows that a g.r. near the equator with either a rising or
a falling frequency can be made self consistent in this way.
Stability
Observation shows that in the mature emission phase wave amplitude and frequency
gradient achieve fairly constant values. This suggests that the g.r. attains a dynamically
stable configuration. At present it is unfortunately not possible to discuss the nature of
such stabilization mechanisms without more detailed knowledge of the g.r. structure.
The question of stability leads naturally to the phenomenon of hook formation. When
the riser frequency reaches about a* - 0.7 power input drops sharpIy. The normal mode
of behaviour in this case is that the wave train terminates. Sometimes a downward hook
may be formed. This process is best regarded as a destabilization of the g.r, due to diminished power supply, and its spontaneous change to a more stable configuration. Note
that such a change in [& can take place with very little or no positional change of the g.r.
The converse process of the upturning of a failer (usually around a* N 0.4) is open to
a similar interpretation.
11. NuMExIcAL
VALUES
We now see how some observed numerical values fit in with the present theory.
The crucial test of the theory will of course be the accurate measurement of the wave
electric fields at the equator. We require values of R* in the range lOa to 1O-5which corresponds to fields of low6 --IV V/m . A few available me~urements, taken at lower
altitudes, correspond reasonably well with these figures.
From the yo* diagram (Fig. 15) we may read off at once the value of 6& which corresponds to an amplitude R* of say OGlOO1, and this is also of the order of OGMOl. This
corresponds to a rate of change of frequency of 7 kc/s, which is quite reasonable.
From the self-consistency relation Equation (55) however, we must also have
t& - oGOOO1- ye?
- y&Xxl
y*-0.01.
We thus see that the resonant particle density, or yo, must be sutEciently great in order for
the observed rates of change of frequency to be produced. The above figure for y. is quite
acceptable for the relative density of particles at these energies (500 ev). It will also give
the large growth rates necessary for a self sustaining g.r. to exist.
12. CONCLUSION
We shall conclude briefly by pointing out some of the flaws in the theory, and with
some suggestions for additional work.
A THEORY OF VLF EMISSIONS
1167
I think the conclusion overall must be that a theory based upon stably trapped second
order resonant particles is able to give a reasonable account of morse triggered VLF
emissions. However the theory is based upon two critical assumptions, and if either of
these proves untenable it will have to be substantially modified. The first assumption is
that k is everywhere parallel to B,,. In fact the question of whistler propagation in the
magnetosphere is very complex and it may well be that a full theory will require the insertion of a perpendicular k component. Secondly we have supposed that the field amplitude is sufficiently great, such that particles may stay trapped for several trapping periods.
If this is not the case the resonant particle currents will have to be computed directly from
particle trajectories.
A great deal of further work remains to be done. A satisfactory explanation of the
sharpness of triggering around cx* = 0.5 was not really given. The structure and stability
of the generation region is not yet understood. Also requiring investigation is the nature
of resonant particle currents at weaker field amplitudes_ A very puzzling problem is
that of the Omega emissions (Kimura, 1968) which appear to trigger at very small amplitudes. Yet another is that of the low frequency triggering (a* - Oql), by natural
whistlers. It really seems that many aspects of the general VLF emission problem are far
from understood. However, it is hoped that this work will at least pave the way for more
sophisticated analyses.
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DAS, A. C, (1968). A
Non linear particle trajectories in a whistler mode wave packet. Imp. Coil. Rep.
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DOWDEN,R. L. (1962). Doppler shifted radiation from electrons a theory of VLF emissions from the
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DUNGEY,J. W. (1968). Waves and particles in the magnetosphere. Physics of the Magnetosphere (Ed.
R. H. Carovillano et al.), p. 218. Reidel.
H~LZZ~ELL,R. A. (1965). Whistlers and Related Ionospheric Phenomena. Stanford University Press,
Stanford, Calif.
HELLIWELL,
R. A. (1967). A theory of discreet VLF emissions from the magnetosphere. J. geophys. Res.
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HOUGHMN,M. J. (1969). The linear theory of wave packets propagating in the whistler mode. J. Plasma
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KE~EL, C. F. and PETXHEK,H. E. (1966). A limit on stably trapped particle fluxes. J.geophys. Res. 71,l.
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SCHRAM,D. C., STRIILAND,W. and ORNSTELN,
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SMITH,R. L. and ANGERAMI,
J. J. (1968). Magnetospheric properties deduced from OGO-1 observations
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Swmr, D. (1968). Particle acceleration by electrostatic waves with spatially varying wave number. J.
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WALTER,F. (1969). Non-ducted VLF propagation in the magnetosphere. Tech. Rep., Stanford Electron.
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NUNN, D. (1970a). Wave particle interactions in an electrostatic wave packet. J. Plasma. Phys. (To be
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