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A z mode electron-cyclotron maser model for bottomside ionospheric harmonic radio emissions

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. A4, PAGES 7017-7026, APRIL 1, 1998
A z mode electronscyclotron maser model for bottomside ionospheric harmonic radio emissions
A. J. Willes and S. D. Bale 1
Astronomy Unit, Queen Mary and Westfield College, University of London, London
Z. Kuncic
Department f Physics nd Astronomy, niversity f Victoria, ritish olumbia, anada
Abstract. A model is proposed or recent ground-based observations of auroral
roar emissions, detected at 2f•e and 3f•e, where f•e is the local electron cyclotron
frequency n the source egion, between 200 and 500 km above the Earth's surface.
Electron cyclotron maser emission is a likely mechanism to account for these
emissions because it naturally produces coherent radiation at harmonics of f•e.
A theory for auroral roar emissions has already been proposed, whereby maser-
generated second X2) and third (X3) harmonic x mode radiation s amplified n
the source egion by multiple reflections off the walls of the density cavity in which
they are produced. After many reflections he X2 and X3 waves propagate along
the density cavity to a ground-based observer. However, it is demonstrated here
with ray-tracing calculations hat it is highly probable hat maser-generated 2 and
X3 radiation is reabsorbed at lower altitudes and thus cannot be detected at the
ground. An indirect maser mechanism s proposed nstead, where maser-generated z mode waves at f•e grow to high levels n the source region and then undergo
repeated nonlinear wave-wave coalescence o produce second- and third-harmonic
waves hat propagate directly to the ground. The z mode waves must satisfy the
necessary kinematic constraints to produce observable second- and third-harmonic
radiation. The dependence of the z mode maser on the temperature and functional
form of the unstable electron distribution is discussed, long with the conditions
required for the coalescence rocesses o proceed and produce the observed evels
of radiation.
1. Introduction
Ground-based observations in the auroral zone of nar-
rowband adiation at twice he (local onospheric) lec-
tron cyclotron requency Kellogg nd Monson, 1979,
1984] have recently been supplemented y the detec-
tion of third-harmonic emissions [Weatherwax t al.,
1993a, 1995]. These "auroral oar" emissions ccur n
the frequency ange 2500-3000 kHz (for the second ar-
monic) and 3800-4300 kHz (third harmonic), nd the
harmonic components can be observed either simulta-
neously or individually. However, when observed si-
multaneously, he two harmonic bands typically do not
exhibit similar eatures or obey an exact harmonic 3:2)
•Now at Space Sciences aboratory, University of Cali-
fornia, Berkeley.
Copyright 1998 by the American Geophysical Union.
Paper number 97JA03601.
0148-0227/98/97JA-03601509.00
frequency elationship Weatherwax t al., 1993a], m-
plying that they can only correspond o second- and
third-harmonic emission f each component s produced
at different source heights (with slightly variant val-
ues of the local electron cyclotron requency ie). A
cyclotron maser theory has been proposed Yoon et
al., 1996] o explain hese observations. n this model
the observed emissions re produced by direct electron-
cyclotron maser emission n the second-harmonic X2)
and third-harmonic X3) magnetoionic modes. The
free energy for the maser is provided by an unstable,
trapped electron population in the lower ionosphere.
Energetic electrons are trapped at these altitudes be-
tween the opposing effects of magnetic mirroring at
lower altitudes and parallel electric fields at higher al-
titudes. Comparison of the relative temporal growth
rates for the fundamental and harmonic x and o modes
reveals hat X2, X3 and O1 (fundamental o mode)
waves can all be produced for typical parameter val-
ues in the bottomside ionosphere. The fundamental x mode is suppressed because the required low densities are not attained in the ionosphere.
7017
7018 WILLES ET AL.' LOWER IONOSPHERIC Z MODE MASER
In this paper we draw attention to growth of funda-
mental z mode waves Z1). The z mode s the "slow"
branch of the magnetoionic extraordinary mode. Fun-
damental z mode waves were not included in the anal-
ysis of Yoon et al. [1996] because hey focused n di-
rect maser emission hat generates waves hat can es-
cape the ionosphere o reach a ground-based bserver.
The z mode waves are usually trapped in the iono-
sphere and can only propagate o the ground n spe-
cial circumstances Horne, 1995]. Previous nalyses f
maser growth rates comparing growth rates in the var-
ious competing modes Melrose et al., 1984; Winglee,
1985; Benson and Wong, 1987] have demonstrated he
dominance of the z mode over fundamental and har-
monic x and o modes over a broad range of parameters,
particularly when the spatial growth rate is considered
(which avors mode waves ecause f their slow group
speed).
Here we argue the case for a z mode-driven maser
mechanism o explain the auroral roar emissions. irst,
we show hat the third-harmonic mode (X3) direct maser spatial growth ates are too low (relative o the
second harmonic) o explain the observed o•nparable
intensities of the two harmonic components. We then
demonstrate, with a ray-tracing code, hat it is highly
probable hat direct, maser-generated 2, X3, and O1
(fundamental o mode) radiation s reabsorbed t lower
altitudes in the source egion. The ray paths of z mode
waves, on the other hand, are restricted o a small range of altitudes about the source height and can thus con- tinue to grow o high evels without encountering trong
absorption. The proposed mechanism is indirect be-
cause he trapped z mode waves must undergo epeated
nonlinear wave coalescences n order to produce second-
and higher-harmonic electromagnetic waves which then
propagate o the ground. A theory for emission t cy-
clotron harmonics due to the repeated coalescence f z
mode waves was first proposed n a model to explain
second-harmonic Melrose nd Dulk, 1984] and higher-
harmonic emission ands [Melrose, 1991] n solar mi-
crowave spike bursts generated n the solar corona. A
later model for solar spike bursts, based on the non-
linear coalescence f electron-cyclotron maser emission
(ECME)-generated electrostatic Bernstein waves, bet-
ter accounts or the observed oninteger atios between the multiple-frequency bands and their comparable n-
tensities Willes and Robinson, 996, 1997]. Indirect
generation mechanisms to produce second- and third-
harmonic auroral roar emissions ave also been pro-
posed by other authors. Weatherwax t al. [1995] uled
out nonlinear wave-wave nteractions etween ower hy-
brid and upper hybrid waves o produce electromag-
netic waves because f the low power produced. A spec-
ulative remark on the possibility of repeated z mode
wave coalescence eing responsible or these emissions
was also made by Weatherwax et al.
The following ections e calculate he spatial growth
rates for the competing wave modes and demonstrate the viability of a z mode maser or plasma parameters relevant o the bottomside onosphere. The conditions
under which z mode growth is favored are discussed,
along with the dependence n the functional orm and
temperature of the unstable electron population. We
then perform ay-tracing calculations n a model density
cavity in the lower ionosphere, or a particular auroral
roar event in which both second- and third-harmonic
emissions are present. We demonstrate both the reab-
sorption of maser-generated 2 waves and the "trap-
ping" of maser-generated z mode waves n the source
region. Finally, we discuss he nonlinear coalescence
processes equired o produce he observed adiation, in
particular, he requirements f energy and momentum conservation nd a comparison f the inferred bright-
ness temperatures with observations.
2. Comparison of Spatial Growth Rates
In this section, yclotron-maser rowth ates are com-
pared or X2, X3, O1, and Z1 waves: We choose ring-
like, hot electron distribution o model he trapped elec- tron population as the source of free energy or maser emission. By comparing he X2 and X3 spatial growth
rates, we argue hat the X3 growth rate is too low to ad-
equately explain observations of simultaneous second-
and third-harmonic auroral roar.
The temporal growth rate of the emitted waves n
mode M, defined so that the energy n the waves grows
as e M(k) , is given y [Melrose, 986]
r(k) f a w(k, , )
c
x +kll f(u) (1)
CU_L U_L • '
where (u) is the distribution unction of electrons nd
the normalized momentum u is defined in terms of the
velocity v and momentum p of the electrons by
p ?v
u - = , (2)
me
and the probability unction wM(k, u, s) satisfies s) -
2•7q•R•(k) =
eor•l•(k)l e•(k)'V(k' ,
x 5 ('),• s• - kllCUll ,
(3)
for the Lorentz actor ? = (1 + u2) /2, RM(k) is the
ratio of electric o total energy for mode M), •vM(k)
is the wave requency, •(k) is the polarization ector, and V(k, u,s) is the velocity unction or a spiraling
charge Melrose, 1986].
The strongest contribution o the growth rate is pro-
vided by the Of(u)/OU_L erm in (1), and positive wave
WILLES ET AL.: LOWER IONOSPHERIC Z MODE MASER 7019
growth thus requires a positive slope in perpendicular
velocities. Omidi et al. [1984] dentified hree possi-
ble sources f free energy n the measured electron dis- tribution in the source region of maser-produced, u-
roral kilometric radiation (at significantly igher alti- tudes han for the auroral oar sources). he two main
sources are a loss cone and a "bump" feature. The
loss cone distribution is produced by magnetic mirror-
ing and therefore should also exist at lower altitudes
(provided hat the source s above he mirror point).
The bump component s composed f electrons rapped
between the upward directed mirror and the down-
ward force on electrons xerted by parallel electric ields
[Louarn t al., 1990]. This is modeled ere by the ring-
like distribution used by Yoon et al. [1996], with
, - a2 52 , (4)
where /• is the pitch angle, and A is the normaliza-
tion actor, with A 1 = •rS/2a Serf(I/5)P(uo/a), or
P(x) - xexp(-x 2) + x/-•(x + 1/2)[1 + err(x)], a is the
thermal spread parameter, u0 defines he perpendicular
offset of the distribution, and 5 defines he pitch angle
width of the distribution.
Yoon et al. [1996] derived a growth ate expression
assuming a ring-like, hot electron distribution and a
thermal background electron population (which con- tributes o damping of waves) nd assuming he semirel-
ativistic approximation to the Lorentz factor, with ? =
I + u2/2. Their calculation ssumed ratio of hot to
thermal electrons f nhot/nth 10 $. In this regime,
the wave dispersion s dominated by the thermal plasma and cold plasma magnetoionic modes may be assumed.
For comparison ith the results of Yoon et al. [1996],
4
-4 -2 0 2 z
Ull
Figure 1. Contour plot (log) of the total electron dis-
tribution unction, omprising hermal electrons C•th
5.8 X 10 $) and a ring-like omponent c• = 0.045, =
0.4, u0 = 2.5c•). Both Ull and uñ (components f u par-
allel and perpendicular o B) axes are scaled n units of
c•. See text for variable definitions.
l0
l0
10 10
l0
10
-4
-5 -6 -7
-8 -9
.... i i i i
0.5 1.0 1.5 2.0
COp/•e
2.5
Figure 2. Comparison f the product of spatial growth
rate Fs = Fc/•evg, and bandwidth Ao•/o• or the X2,
X3, Z1, and O1 wave modes, maximized over wave re-
quency nd angle, as a function of
we assume the same electron distribution, with similar
parameters: C•hot 0.06 (2 keV electrons), 5 = 0.4,
u0 = 2.5c•, nd C•th - 5.8 X 10 4 (8.6 X 10 2 eV elec-
trons). For the Maxwellian component, o 0 and the
limit 5 -+ oc is taken. The total electron distribution
(thermal and trapped) is shown n Figure I (with an
exaggerated hermal temperature or clarity).
The growth rates for each mode vary most markedly
with the ratio of frequencies, •p/•e. Figure 2 shows he
product of the spatial growth rate (temporal growth rate divided by group speed of waves) and the band- width Ao•/o•, maximized over wave requency nd an-
gle, as a function of o•p/•e. The product of the spa-
tial growth rate and the bandwidth gives he best mea-
sure of the amplification rate for when the maser does
not saturate and the electron cyclotrbn requency aries
with height [Hewitt et al., 1983; Melrose et al., 1984;
Melrose, 1991]. The spatial growth is more relevant
in this context because mode waves have significantly
lower group speeds han X2, X3, or O1 waves where he
group speed s close o c). This is important f growth
occurs n small, localized regions or if the maser oper-
ates close o a state of marginal stability, where growth
rates are small and the maser does not saturate Robin-
son, 1991; Melrose, 991], so that the z mode waves
spend more time in the growth region and hence expe- rience more amplification. It is also necessary o take
into account the bandwidth because he local cyclotron
frequency varies with altitude and the waves can move
out of the region n which they can grow. Additionally,
wave absorption rom the background Maxwellian is in-
cluded n the growth ate calculation. or o•p/•e • 0.1,
the fundamental x mode has the highest growth rate
(not shown n Figure 2). However, n the bottom-
side onosphere, nly values or o•p/•e above 0.3 are observed Benson nd Wong, 1987; Yoon et al., 1996].
Figure 2 shows hat Z1, O1, and X2 waves have com-
parable growth ates or 0.3 • o•p/•e • 0.8. The reason
7020 WILLES ET AL.: LOWER IONOSPHERIC Z MODE MASER
for the discontinuity n the slope of. he Z1 curve near
wp/f•e • I is discussed n Section . Weatherwax t al.
[1995] showed hat all these modes have growth ates
that exceed the collisional damping rate in the lower ionosphere. Results similar to Figure 2 are obtained if a loss cone electron distribution is assumed n place
of the ring distribution. In both cases he X3 spatial growth rate is at least 2 to 3 orders of magnitude be- low the X2 growth rate. Ray-tracing calculations f the path of propagation of maser-emitted X2 and X3 waves
in a model cavity density structure Yoon et al., 1996]
predict comparable ath lengths to within a factor of
2) for the propagating 2 and X3 waves n the source e-
gion. It is thus unlikely that the third-harmonic compo-
nent of auroral emission s produced by direct X3 emis- sion because t is often observed at comparable ntensi-
ties to the second-harmonic omponent Weatherwax t al., 1995], whereas he theory suggests hat the product
of spatial growth rate and path length is considerably lower for X3 than X2 waves. The X2 spatial growth rate
is itself quite low, and many passes hrough the source
region (after reflections rom the cavity walls) are re- quired o produce everal -foldings f growth Yoon et al., 1996].
If the assumption hot/nth << I is not valid in the
source egion, then the X2 and O1 spatial growth rates
are significantly igher relative o the Z1 and X3 rates)
owing to slower group speeds s a result of splitting of
the dispersion elation [Winglee, 1985]. This further
widens the gap between he spatial growth rates of the
X2 and X3 waves.
3. Ray Tracing of Maser-Generated
Waves
In this section we report on the results of ray-tracing calculations, modeling a particular auroral roar event n
which simultaneous second- and third-harmonic com-
ponents were observed. We first show that X2 rays
encounter a strong absorption region just below their source altitude and thus cannot propagate o a ground-
based observer. This conclusion extends also to X3 and
O1 waves. We then demonstrate that, for the appro-
priate parameters, z mode waves can be trapped in the
source egion and thus grow to high levels before being
damped. These z mode waves can coalesce o produce
second- and third-harmonic electromagnetic waves, as
discussed in section 4.
The numerical ray-tracing calculations discussed n
this section are based on the geometric optics approxi-
mation, which is valid in our model because he wave-
length of the propagating waves s well below he other
spatial scales n the source egion. It is possible, how-
ever, that in the auroral ionosphere, smaller spatial
Linear Scale Black = 50 White = 5
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.'.•.'" .'.'.•.'•.'.'•:½.'.•.....:•'*,,•.... ..•
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•""•':':•' :•
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401•0 /UT
Figure 3. Dynamic spectrum for an auroral roar event with both second- and third-harmonic
components from Weatherwax t al., 1993a,b].
WILLES ET AL.: LOWER IONOSPHERIC Z MODE MASER 7021
structures may exist where he geometric ptics approx-
imation is not valid. The ray equations are
dr 0w(k, r)
X = ok ' (5) 2
dk Ow(k, )
dt Or ' (6)
where t is a parametric epresentation f the points
along he wave path, defined o hat the tangent o the
ray path s equal o the group elocity, ith va(k r) :=
&o(k, r)/Ok. The ray-tracing ode sed ere s an adap-
tation of an earlier code ormulated o study he prop-
agation and absorption ffects of maser-generated un-
damental and o mode adiation n a coronal lux oop
[Kuncic nd Robinson, 993].
In our ray-tracing analysis we choose o model a par-
ticular auroral roar event (May 11, 1992, between 020
and 1030 UT [Weatherwax t al., 1993a, , Figure ])
that exhibits simultaneous second and third harmonics.
The dynamic pectrum or this event s shown n Fig-
ure 3. Assuming ipolar variation n the magnetic ield
strength and that the two components re emitted at
twice and three times he local electron yclotron re-
quency, he second-harmonic omponent 2.75 MHz) is
emitted approximately t a height of 200 km, and the
third harmonic 3.9 MHz) is emitted t a height f 300
km. It then follows hat the source egion where he
trapped energetic lectrons xist) extends t least 100
km in the vertical direction. A schematic epresenta-
tion of the source egion for auroral roar emissions s
shown n Figure 4.
Z /
400 km -
300 km - 200 km - 100 km -
Density cavity
X X
Source eight of 3•eemission
Source eight of 2f• eemission
B
Ground )
x
Figure 4. Schematic icture of the auroral roar source
region, with the second- nd third-harmonic ompo-
nents generated t different ltitudes n a field-aligned
density cavity.
(--.0 /• e
-150
.... i , , i , , i .... .... i , , i , I I I ....
-100 -50 0 50 100 150
x (km)
Figure 5. Model density cavity profile, using 7) with
the following arameters: - 0.65, b = 3, W = 6 x 104 m, andA=1.2x104m. In the lower onosphere he ratio wp/fie exceeds nity
(typically -5). For such igh alues f wp/fie, one f
the X2, O 1, or Z1 modes s unstable, s shown n Figure
2. However, ower values f cop/fie xist n field-aligned density cavities, where wp/fie can be low enough or
any of the X2, X3, O1, or Z1 waves o grow. In the
model of Yoon et al. [1996], such cavities also act as
a waveguide, allowing X2 and X3 waves o propagate
to the ground. We adopt a model similar o that of
Yoon et al. [1996] or the density cavity profile, based
on ground-based ncoherent scatter radar observations
[Doe et al., 1993, 1995], with
cop- 2 tanh A +b2_a i (7)
:
where b is the value of cop utside he cavity (which
varies ith he onospheric ensity rofile; ee igure ),
a is the value of cop t the center of the cavity which
remains constant), w is the width of the cavity, and
A determines he slope f the cavity. Equation 7) is
only well defined or b > a. Figure 5 shows typical
density profile or b = 3, a = 0.65, w - 6 x 104 m,
and A - 1.2 x 104 m. In addition, e ncorporate nto
our model he variation f he ocal yclotron requency,
which slowly ncreases ith decreasing ltitude, and he
variation of the plasma requency ith height which
determines arameter b in (7)) using he international
reference onosphere Bilitza et al., 1993] or the time
and coordinates f the Weatherwax t al. [1993a] vent.
Figure displays he variation f electron ensity ith
height sed n our ay-tracing odel. n the ray-tracing
calculations resented elow we assume hat the hot,
trapped electron distribution only exists in the inner
region f the density avity, or ]x _< x 104 m.
3.1. X2 Waves
Figure 7 shows ontours f constant rowth solid
contours) nd absorption dashed ontours) ate for
maser emission n the second-harmonic mode, assum-

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