Ann. Geophys., 27, 2711–2720, 2009
www.ann-geophys.net/27/2711/2009/
© Author(s) 2009. This work is distributed under
the Creative Commons Attribution 3.0 License.
Annales
Geophysicae
Artificial E-region field-aligned plasma irregularities generated at
pump frequencies near the second electron gyroharmonic
D. L. Hysell and E. Nossa
Earth and Atmospheric Science, Cornell University, Ithaca, NY, USA
Received: 2 March 2009 – Revised: 15 June 2009 – Accepted: 23 June 2009 – Published: 7 July 2009
Abstract. E region ionospheric modification experiments
have been performed at HAARP using pump frequencies
about 50 kHz above and below the second electron gyroharmonic frequency. Artificial E region field-aligned plasma
density irregularities (FAIs) were created and observed using the imaging coherent scatter radar near Homer, Alaska.
Echoes from FAIs generated with pump frequencies above
and below 2e did not appear to differ significantly in experiments conducted on summer afternoons in 2008, and the resonance instability seemed to be at work in either case. We argue that upper hybrid wave trapping and resonance instability at pump frequencies below the second electron gyroharmonic frequency are permitted theoretically when the effects
of finite parallel wavenumbers are considered. Echoes from a
sporadic E layer were observed to be somewhat weaker when
the pump frequency was 50 kHz below the second electron
gyroharmonic frequency. This may indicate that finite parallel wavenumbers are inconsistent with wave trapping in thin
sporadic E ionization layers.
Keywords. Radio science (Waves in plasma) – Space
plasma physics (Active perturbation experiments; Waves and
instabilities)
1
Introduction
The production of magnetic field-aligned plasma density irregularities (FAIs) is a signature feature of RF ionospheric
modification experiments. Visible to coherent scatter radars,
the irregularities provide a means of diagnosing the experiments with ground-based remote sensing. A number of
radar observations have been conducted in concert with ionospheric heating experiments in an attempt to elucidate the
Correspondence to: D. L. Hysell
([email protected])
mechanisms responsible for creating the FAIs and other related phenomena (see reviews by Robinson, 1989; Frolov
et al., 1997; Gurevich, 2007). While questions remain, the
finding that FAIs are produced near the upper hybrid resonance level in the plasma rather than at the critical height
(e.g. Djuth et al., 1985; Noble et al., 1987) focuses attention on two closely related instability processes. One of
these is the thermal parametric instability (Grach et al., 1978;
Das and Fejer, 1979; Fejer, 1979; Kuo and Lee, 1982; Dysthe et al., 1983; Mjølhus, 1990), and the other is the resonance instability (Vas’kov and Gurevich, 1977; Inhester
et al., 1981; Grach et al., 1981; Dysthe et al., 1982; Lee and
Kuo, 1983; Mjølhus, 1993).
In the thermal parametric instability, or the thermal oscillating two-stream instability (OTSI) more precisely, the
pump electromagnetic wave is resonantly converted in the
presence of field-aligned plasma density irregularities into
upper hybrid waves which heat the plasma collisionally and,
in turn, intensify the irregularities through differential thermal forcing. If the amplitude of the irregularities (striations) becomes sufficiently large, upper hybrid waves may
be trapped within them, causing additional mode conversion,
heating, and trapping in tandem with the formation of thermal cavitons. This is the condition for the resonance instability, which is characterized by explosive wave growth. Coherent scatter is then indicative of the meter-scale striations
(e.g. Stubbe, 1996).
The behavior of upper-hybrid waves depends strongly on
the frequency with respect to harmonics of the electron gyrofrequency, ne , mainly because of sharply increased collisionless damping at these frequencies. A number of closely
related phenomena, including stimulated electromagnetic
emission, electron and ion heating, optical emissions, and irregularity generation are also affected or interrupted when
pumping at or near gyroharmonic frequencies (see Kosch
et al., 2005; Mishin et al., 2005 for experimental and theoretical summaries). By investigating the kinetic dispersion
Published by Copernicus Publications on behalf of the European Geosciences Union.
2712
relation for upper hybrid/ electron Bernstein waves, Mjølhus
(1993) argued that striation formation should be suppressed
at heating frequencies near ne for n ≥ 3. Huang and Kuo
(1994), Istomin and Leyser (2003), and Gurevich (2007)
reached similar conclusions and also predicted that the suppression should be asymmetric, extending to frequencies further below the gyroharmonic frequency than above it. Direct
evidence for this phenomenon in the F region came first in
the form of reduced anomalous absorption and subsequent
intensification of processes occurring above the upper hybrid
resonance height (Honary et al., 1999) and then in the form
of reduced coherent scatter measured at Sura and by SuperDARN (Ponomarenko et al., 1999; Kosch et al., 2002). Additional, albeit indirect, evidence came from optical observations of artificial airglow and inferences about upper-hybrid
wave-induced electron acceleration (e.g. Kosch et al., 2005).
The situation for pump frequencies near the second electron gyroharmonic appears to be more complicated and could
not be explored experimentally from the mid-1980s until recently with the advent of a low-frequency heating capability at HAARP. Coherent scatter is observed for pump frequencies near 2e , and enhancements have been reported
at frequencies just above 2e (Fialer, 1974; Minkoff et al.,
1974; Kosch et al., 2007). Airglow intensifications are also
reliably observed for pump frequencies slightly above 2e
(Haslett and Megill, 1974; Djuth et al., 2005; Kosch et al.,
2005). On the theoretical side, the threshold for excitation of thermal parametric instability can be shown to be
reduced in the vicinity of the second electron gyroharmonic
frequency (Grach, 1979), explaining the observed, aforementioned overshoot.
However, analysis also suggests that wave trapping may
be prohibited entirely at pump frequencies below the second
electron gyroharmonic frequency (Mjølhus, 1993; Huang
and Kuo, 1994). The prediction has especially important implications for artificial E region FAIs, which are routinely
and often necessarily (in view of the low E-region critical
frequency) generated using pump frequencies below and often well below 2e (e.g. Fialer, 1974; Hysell, 2008). For example, the FAIs observed by Nossa et al. (2009) were mainly
generated using a pump frequency of just 2.75 MHz. Theory
in this area has mainly concentrated on F region applications,
but renewed interest in E region heating experiments prompts
reinvestigation of the experimental record and the underlying
physics.
In this paper, we investigate the generation of FAIs in the
E region ionosphere at pump frequencies above and below
the second electron gyroharmonic. The frequencies used are
insufficiently close (e.g. see Gurevich, 2007) to 2e to evaluate upper hybrid wave suppression and asymmetry and are
instead widely separated to ensure bracketing of the second
gyroharmonic under a variety of E region conditions. E region experiments offer some practical advantages from the
diagnostic point of view. Using VHF radar, high-latitude artificial E region FAIs can be observed with relatively little
Ann. Geophys., 27, 2711–2720, 2009
D. L. Hysell and E. Nossa: FAI generation near 2e
reliance on refraction to satisfy the Bragg scattering condition and correspondingly little ambiguity with regard to
echolocation. Likewise, E region echo Doppler shifts are
generally small enough to render the target underspread, permitting unambiguous spectral estimation using conventional
pulse-to-pulse signal processing. Furthermore, the daytime E
region ionosphere is relatively consistent and invariant on a
day-to-day and minute-to-minute basis by comparison to the
F region, affording more possibilities for controlled experiments. However, the critical frequency in the high-latitude E
region is seldom more than about 3 MHz, making it difficult
to explore gyroharmonics with n ≥ 3. Sporadic E layers can
be utilized for observations at higher pump frequencies, but
only at the expense of the controlled experiment aspect of the
observations.
2
Observations
The coherent scatter radar used for this investigation is located at the NOAA Kasitsna Bay Laboratory near Seldovia,
Alaska, approximately 470 km southwest of HAARP. It operates at 29.8 MHz using a 12 kW peak power transmitter
and two side-by-side transmitting antenna arrays with a combined gain of approximately 18.5 dBi. For the experiments
described here, 13-bit Barker-coded pulses with a 10 µs baud
width and a pulse repetition frequency (PRF) of 405 Hz were
used, giving a transmitter duty cycle of 5.3% and a maximum unambiguously resolvable Doppler shift of just over
1000 m/s. Reception is performed using 6 spaced antenna
arrays, providing 15 approximately coplanar, nonredundant
baselines for interferometry. Additional details about the system and some of the diagnostics it can perform were given by
Hysell (2008).
The nominal electron gyrofrequency, e , in the E region
over HAARP is precisely 1.5 MHz (∼1.511 MHz at 105 km
altitude according to IGRF-2005). The E region critical frequency meanwhile typically reached 3 MHz at midday in
the summer of 2008, with an associated peak upper-hybrid
frequency of 3.35 MHz. This made it possible to use heating frequencies above and below the second electron gyroharmonic frequency, 2e , during the summer months, even
during solar minimum, with the expectation of meeting the
upper-hybrid resonance condition in the E region.
Attempts to generate E region field-aligned plasma density irregularities over HAARP using O-mode zenith heating
at pump frequencies about 50 kHz above and below the second electron gyroharmonic frequency were made at midday
on 28 and 29 July 2008, during the Polar Aeronomy and Radio Science (PARS) Summer School period. The frequencies
2.945 MHz and 3.045 MHz were available for these experiments. Heating was performed using the full power of the
HAARP facility with zenith pointing and CW modulation
following a 1-min on, 1-min off pattern. Results for 28 July
are shown in Fig. 1.
www.ann-geophys.net/27/2711/2009/
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D. L. Hysell and E. Nossa: FAI generation near 2e
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0
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10
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0
100
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200
0
-5
-10
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Fig. 1. Range-time Doppler intensity (RTDI) plot of backscatter from artificial E region FAIs over HAARP observed on 28 July 2008. Here,
Fig.
Range-time
intensityof(RTDI)
plot of
backscatter
from artificial ratio
E region
FAIsDoppler
over HAARP
observed
on width.
July 28,Note
2008.that
Here,
the 1.
brightness,
hue,Doppler
and saturation
the pixels
denote
echo signal-to-noise
(SNR),
shift, and
spectral
the
the
brightness,
hue, and saturation
of the
pixels
denote
(SNR),
Doppler
shift, andrange
spectral
width.
the
echoes
from heater-induced
FAIs are
range
aliased
andecho
that signal-to-noise
their true rangeratio
is greater
than
their apparent
by 370
km. Note
The that
average
echoes
from heater-induced
FAIs are
range
aliased80–140
and that
true range
is greater
thanplot.
theirEchoes
apparent
range
by 370
km.areThe
signal-to-noise
ratio for apparent
ranges
between
kmtheir
is plotted
beneath
the RTDI
from
meteor
trails
alsoaverage
visible
signal-to-noise
ratioincoherent
for apparent
ranges between
kmisisabout
plotted
the RTDI
plot.
frommeteor
meteorechoes
trails are
in the figure. The
integration
time for80–140
the figure
3 s.beneath
Note that
specular
andEchoes
long-lived
are also
also visible
visible
inthroughout
the figure.the
The
incoherent integration time for the figure is about 3 s. Note that specular and long-lived meteor echoes are also visible
figure.
throughout the figure.
In Fig. 1, Homer radar data are portrayed in range-timetral
analysis. The echoes
Doppler
shiftsthe
of brightness,
about -50
Doppler-intensity
(RTDI) have
format,
in which
m/s
andof
RMS
of about
50 m/s in this
hue,(red
andshifts)
intensity
the half-widths
pixels represent
the signal-to-noise
case.
lowershift,
paneland
of the
figurewidth
shows
signalratio,The
Doppler
spectral
ofthe
theaverage
echoes accordto-noise
of echoes
km apparent
range
ing to theratio
legend
shown.between
Coherent80–140
integration
by a factor
of 4
(450–510
kmtotrue
range).
Thetoincoherent
integration
was applied
these
data prior
detection and
spectral time
analfor
theThe
dataechoes
in the figure
is approximately
3 s. −50 m/s (red
ysis.
have Doppler
shifts of about
shifts) and RMS half-widths of about 50 m/s in this case. The
For this
experiment,
the shows
pump frequency
2.945 MHz
lower
panel
of the figure
the averagewas
signal-to-noise
(3.045
MHz)
during
the
1st,
3rd,
5th,
and
7th
(2nd,
4th,(450–
and
ratio of echoes between 80–140 km apparent range
6th)
heating
events
depicted
in
Figure
1.
The
echoes
asso510 km true range). The incoherent integration time for the
ciated
lower
heating frequency
data inwith
the the
figure
is approximately
3 s. are slightly stronger
(<1 dB) than at the higher frequency, despite the fact that the
HAARP
antenna
has slightly
higher
gain (∼0.3
dB) atMHz
the
For this
experiment,
the pump
frequency
was 2.945
higher
The
for5th,
the and
lower
(3.045frequency.
MHz) during
theechoes
1st, 3rd,
7thfrequency
(2nd, 4th,also
and
appear
to fall events
at slightly
shorter
ranges
the higher
1. than
The for
echoes
associ6th) heating
depicted
in Fig.
frequency,
that could
be indicative
of a small
difated with something
the lower heating
frequency
are slightly
stronger
ference in scattering altitude. The Doppler spectra for the
www.ann-geophys.net/27/2711/2009/
(<1 dB) than at the higher frequency, despite the fact that the
two
heating
frequencies
are indistinguishable.
HAARP
antenna
has slightly
higher gain (∼0.3 dB) at the
higher frequency. The echoes for the lower frequency also
Data from
segment
of anranges
identical
perappear
to falla short
at slightly
shorter
thanexperiment
for the higher
formed
on
July
29,
2008,
are
shown
in
Figure
2
(the
complete
frequency, something that could be indicative of a small difdataset,
by this
short segment,
appears
in Figure
1
ference typified
in scattering
altitude.
The Doppler
spectra
for the
of
Nossa
et
al.
(2009)).
The
echoes
received
this
time
had
two heating frequencies are indistinguishable.
very small Doppler shifts, and the data in the figure were
coherently
integrated
by a factor
16 priorexperiment
to detectionperto
Data from
a short segment
of anofidentical
improve
the29
overall
ratio.
The2pump
frequen(the complete
formed on
July signal-to-noise
2008, are shown
in Fig.
cies
for the
first by
andthis
second
event
werein2.945
dataset,
typified
shortheating
segment,
appears
Fig. 1and
of
3.045
As in received
the previous
example,
the
NossaMHz,
et al., respectively.
2009). The echoes
this time
had very
echoes
observedshifts,
here were
at thewere
lowercoherheatsmall Doppler
and slightly
the datastronger
in the figure
ing
frequency.
thisoftrend
cannot
extrapolated;
ently
integratedNote
by athat
factor
16 prior
to be
detection
to imheating
experiments
conducted justratio.
prior to
those
shown
here
prove the
overall signal-to-noise
The
pump
frequenat
a frequency
of and
2.75 second
MHz produced
coherent
cies
for the first
heating event
wereechoes
2.945with
and
signal-to-noise
ratios comparable
generated
at 2.945
3.045 MHz, respectively.
As in to
thethose
previous
example,
the
MHz. Echoes from the higher heating frequency also extend
Ann. Geophys., 27, 2711–2720, 2009
D. L. Hysell and E. Nossa: FAI generation near 2e
: FAI GENERATION NEAR 2Ωe
100
50
Range (km)
150
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UT (hrs) Tue Jul 29 2008
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10
-60
-40
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-20
0
20
v (m/s)
40
60
5
0
-10
-5
SNR (dB)
10
-10
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UT (hrs) Tue Jul 29 2008
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20:53:00
Fig. 2. RTDI plot of backscatter from artificial E region FAIs over HAARP observed on 29 July 2008. The incoherent integration time for
Fig.
2. RTDI
plot of
backscatter
from artificial E region FAIs over HAARP observed on July 29, 2008. The incoherent integration time for
the figure
is about
0.158
s.
the figure is about 0.158 s.
echoes observed here were slightly stronger at the lower heatover
a somewhatNote
broader
spantrend
of ranges
echoes from
ing frequency.
that this
cannotthan
be extrapolated;
the
lowerexperiments
frequency, something
of a
heating
conductedthat
justcould
priorbetoindicative
those shown
larger
volume
and aMHz
higher
scattering
altitudeechoes
at the
here atmodified
a frequency
of 2.75
produced
coherent
higher
frequency. Close
relatively
high
with signal-to-noise
ratiosexamination
comparable of
to the
those
generated
at
time
data in
Figure
2 shows
that the
rise andalso
fall
2.945resolution
MHz. Echoes
from
the higher
heating
frequency
times
theaechoes
at the
two heating
like
extendfor
over
somewhat
broader
span offrequencies
ranges thanare,
echoes
the
Doppler
spectra,
indistinguishable.
from
the lower
frequency,
something that could be indicative
of a larger modified volume and a higher scattering altitude
an experiment
similar to theofpreceding
two
at Additionally,
the higher frequency.
Close examination
the relatively
was
on February
29,2 2008,
priorthe
torise
local
high performed
time resolution
data in Fig.
shows that
andmidfall
night
conditions
a dense
times during
for the active
echoesauroral
at the two
heating when
frequencies
are,(critilike
cal
FoEs ∼
4 MHz), blanketing sporadic E layer
the frequency
Doppler spectra,
indistinguishable.
was over HAARP. Figure 3 shows the results in RTDI forAdditionally,
an experiment
similar
to theafter
preceding
mat.
Natural auroral
echoes are
evident
about two
815
was
performed
on
29
February
2008,
prior
to
local
midUT. The riometer at HAARP indicated significant absorpnightafter
during
active
auroral
conditions
when a dense
tion
about
800 UT
but little
in the preceding
hour.(critiVery
cal
frequency
FoEs
∼4
MHz),
blanketing
sporadic
E layer
weak echoes from artificially generated FAIs were also
dewas over
HAARP.
Figure
3 shows the
results
in RTDI
tected.
These
echoes
had relatively
large
Doppler
shiftsforof
mat. Natural
auroral
abouthalf
08:15
UT.
about
150 m/s
(blueechoes
shifts)are
andevident
RMS after
spectral
widths
Theabout
riometer
HAARP
significant
absorption
afof
100 at
m/s.
As inindicated
the previous
examples,
the heating frequency alternated between 2.945 and 3.045 MHz, with
Ann. Geophys., 27, 2711–2720, 2009
ter about 08:00 UT but little in the preceding hour. Very weak
minute-long
intervals
at theFAIs
higher
frequency
beginechoes
from heating
artificially
generated
were
also detected.
ning atechoes
744, 748,
756, 800,
andDoppler
804 UT and
interleaving
These
had752,
relatively
large
shifts
of about
lower-frequency
heating
In this
case,
the echoes
re150
m/s (blue shifts)
and intervals.
RMS spectral
half
widths
of about
ceived
strongerexamples,
at the higher
heating frequency
frequency
100
m/s.appear
As in to
thebeprevious
the heating
by about 3–5
dB. Low
intensity
and
significant
alternated
between
2.945backscatter
and 3.045 MHz,
with
minute-long
background
variability
both undermine
clarity of
reheating
intervals
at the higher
frequencythe
beginning
at this
07:44,
sult to some
07:48,
07:52,extent,
07:56,however.
08:00, and 08:04 UT and interleaving
lower-frequency heating intervals. In this case, the echoes received appear to be stronger at the higher heating frequency
3 about
Analysis
by
3–5 dB. Low backscatter intensity and significant
background variability both undermine the clarity of this reThe to
most
significant
finding from the July, 2008 experiments
sult
some
extent, however.
described above is that there appear to be no significant
differences between FAIs generated with pump frequencies
50 kHz above and below the second electron gyrohar3about
Analysis
monic frequency. Small changes in echo range and intensity
The
significant
2008ofexperiweremost
reported,
but thesefinding
are no from
larger the
thanJuly
the kind
secular
ments
described
above
is
that
there
appear
to
no signifvariations routinely observed and attributed tobebackground
icant
differences
between and
FAIsthegenerated
pump
fredensity
profile variability
sensitivitywith
of the
thermal
quencies
about
50
kHz
above
and
below
the
second
electron
oscillating two-stream instability to the height of the upper
hybrid resonance layer with respect to the pump standing
www.ann-geophys.net/27/2711/2009/
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D. L. Hysell and E. Nossa: FAI generation near 2e
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0
-200
-100
0
100
v (m/s)
200
-5
-10
-15
SNR (dB)
0
-15
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8:00
UT (hrs) Fri Feb 29 2008
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7:30
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UT (hrs) Fri Feb 29 2008
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8:30
Fig. 3. RTDI plot of backscatter from artificial E region FAIs over HAARP observed on 29 February 2008, during a sporadic E layer event.
Fig.
3. RTDI
plotechoes
of backscatter
from
regionoutside
FAIs over
HAARPkm
observed
on February
29, 2008,
duringtime
a sporadic
E layerisevent.
Natural
auroral
are evident
inartificial
apparentEranges
the 90–140
interval.The
incoherent
integration
for the figure
about
Natural
13 s. auroral echoes are evident in apparent ranges outside the 90–140 km interval.The incoherent integration time for the figure is about
13 s.
gyroharmonic frequency. Small changes in echo range and
wave.
Other
diagnostics
unavailable
these than
experiments
intensity
were
reported, but
these are for
no larger
the kind
may
do
so,
but
the
coherent
scatter
radar
does
not clearlyto
of secular variations routinely observed and attributed
distinguish
generated
with
pump frebackgroundbetween
densityirregularities
profile variability
and the
sensitivity
of
quencies
above
and
below
2Ω
.
As
hysteresis
effects
indicae
the thermal oscillating two-stream
instability to the height
of
tive
wavehybrid
trapping
are evident
experiments
the of
upper
resonance
layer in
with
respect to involving
the pump
pump
frequencies
below
2Ω
(Nossa
et
al., 2009),
theory
standing wave. Other diagnostics
unavailable
for these
exe
governing
wave
trapping
in
this
frequency
regime
warrants
periments may do so, but the coherent scatter radar does
additional
investigation.
not clearly
distinguish between irregularities generated with
pump
frequencies
above
and and
below
2e . and
As hysteresis
efFollowing
McBride
(1970)
Huang
Kuo (1994),
fects
indicative
wave of
trapping
are evident
in in
experiments
we
seek
a kineticoftheory
upper hybrid
waves
a weakly
et al.,
2009),
involving
pumpthat
frequencies
2e (Nossa
ionized
plasma
includesbelow
the effects
of finite
parallel
theory governing
wave trapping
in this for
frequency
regime
wavenumbers.
An appropriate
expression
the longitudiwarrants
additional
investigation.
nal
dielectric
permittivity
for a magnetized Maxwellian electron plasma incorporating the BGK collisional integral has
Following McBride (1970) and Huang and Kuo (1994),
been derived by Alexandrov et al. (1984) (see also Mjølhus
we seek a kinetic theory of upper hybrid waves in a weakly
(1993) and Kryshtal (1998) for notational guides):
ionized plasma that includes the effects of finite parallel
wavenumbers. An appropriate expression for the longitudinal dielectric permittivity for a magnetized Maxwellian elecǫ(ω, k)
www.ann-geophys.net/27/2711/2009/
tronk plasma
incorporating the BGK collisional integral has
i kj
ǫij (ω,by
k)Alexandrov et al. (1984) (see also Mjølhus
(1)
=
been 2derived
k
)
(1993) and(Kryshtal
(1998) for
notational
guides):
P∞
ω+iν
2
kd2 1 + n=−∞ ω+iν−nΩ [W (αn ) − 1] Λn (χ )
P∞
=
1+ 2
iν
(ω,k)
k
[W (αn ) − 1] Λn (χ2 )
1 + n=−∞ ω+iν−nΩ
e
ki kj
= 2 ij (ω,k)
(1)
with
k
(
)
p
P
∞
ω+iν
1+
(αn ) − 1]3n (χ 2 )
[W
αn =k 2(ω +
iν −n=−∞
nΩe )/(|k
/m)
k| T
ω+iν−n
= 1 + d2
P
iν
2
1 + −z ∞
Λn (z) =k In (z)e
n=−∞ ω+iν−ne [W (αn ) − 1]3n (χ )
2
χ2 = k⊥
KT /mΩ2e
with
2 2
= k⊥
ρ
p
αn = (ω + iν − ne )/(|kk | T /m)
where Ωe is the−z
electron gyrofrequency, ν is the electron3n (z) =
In (z)e frequency, k⊥ and kk are the perpendicular
neutral
collision
2
2
= k⊥
KT /m2e
and χ
parallel
wavenumbers,
respectively, kd ≡ λ−1
d is the De2 2 ρ is the electron gyroradius, and where the
bye wavenumber,
= k⊥ ρ
other symbols have the usual meaning. Also, In is the modiwhere
e function
is the electron
gyrofrequency,
ν n
is and
the W
electronfied
Bessel
of the first
kind of order
is the
neutral
collision
frequency,
k
and
k
are
the
perpendicular
plasma dispersion function, which
is krelated to the complex
⊥
error function (e.g. Fried and Conte (1961)).
Ann. Geophys., 27, 2711–2720, 2009
2716
6
Fig. 4.
upper
hybrid/electron
Bernstein
4. Dispersion
Dispersioncurves
curvesforfor
upper
hybrid/electron
Bern−1,, and
waveswaves
versusversus
k⊥ with
1.5 MHz,
ν =ν=5000
5000 s−1
e /2π
stein
k⊥ Ω
with
e=/2π=1.5
MHz,
and
= 2π/10
m−1forforsixsixdifferent
different plasma
plasma frequencies
frequencies (ω
kkkk=2π/10
m−1
(ωpp/2π
/2π ∈∈
{1.9, 2.2, 2.5, 2.8, 3.1, 3.4}
MHz).TheThe
corresponding
upper
hy{1.9,2.2,2.5,2.8,3.1,3.4}
MHz).
corresponding
upper
hybrid
brid
frequencies,
ω
/2π,
are
indicated
by
+
symbols
on
the
ordiuh
frequencies, ωuh /2π , are indicated by + symbols on the ordinate.
nate. Dispersion
corresponding
to upper
frequencies
Dispersion
curvescurves
corresponding
to upper
hybridhybrid
frequencies
below
below (above)
2Ω
3 MHz
are indicated
with
black
solid
(red
e /2π
(above)
2e /2π
=3
MHz= are
indicated
with black
solid
(red
dashed)
dashed) lines.
lines.
andThe
parallel
respectively,
kd ≡with
λ−1
roots wavenumbers,
of (1) can be found
numerically
thethe
aidDeof
d is
bye
wavenumber,
ρ is the
electron gyroradius,
and where
the
a few
readily available
computational
tools. These
include
other
symbols
havedispersion
the usual meaning.
Also, In isathe
modian efficient
plasma
function calculator,
modified
fied
Bessel
function
of the and
first akind
of order
n and Wvariant
is the
Bessel
function
calculator,
globally
convergent
plasma
dispersion
function,
is related
to Amos,
the complex
of Newton’s
method
(Poppe which
and Wijers,
1990;
1986;
error
function
1961).
Powell,
1970).(e.g.
NoteFried
that and
the Conte,
derivatives
of (1) necessary to
The roots
Eq. (1)matrix
can befor
found
numerically
withelementhe aid
populate
the of
Jacobian
the root
solver have
of
a few
readily
available computational tools. These include
tary
analytic
forms.
an The
efficient
function
calculator,
a modified
rootsplasma
of (1) dispersion
for frequencies
in the
vicinity of
the secBessel
function
calculator, and
a globally
convergent
ond electron
gyroharmonic
frequency
plotted
versusvariant
transof
Newton’s
methodk⊥(Poppe
and Wijers,
1990;
Amos,
verse
wavenumber
are shown
in Figure
4. For
the 1986;
comPowell,
Note
that Ω
the
derivatives
of Eq.ν (1)
necessary
putation,1970).
we have
taken
= 1.5 MHz,
= 5000
s−1 ,
e /2π
−1
to
root solver
have
elekk populate
= 2π/10the
mJacobian
and Tmatrix
= 350forK.the
Curves
for six
differmentary
analytic
forms. equally spaced between 1.9 and 3.4
ent plasma
frequencies,
The are
roots
of Eq.
for frequencies
in hybrid
the vicinity
of
MHz,
plotted.
The(1)corresponding
upper
frequenthe
electron
gyroharmonic
frequency
versus
cies,second
ωuh /2π,
are indicated
by + symbols
onplotted
the ordinate.
4. For
the
transverse
wavenumber
k⊥ to
areconditions
shown inwith
Fig.upper
Three curves
corresponding
hybrid
computation,
we have
taken
MHz,gyroharmonic
ν=5000 s−1 ,
frequencies below
(above)
thesecond
electron
e /2π=1.5
kfrequency
and T =350
Curves
for six
different
plasma
are−1plotted
usingK.
dashed
black
(solid
red) lines.
k =2π/10 m
frequencies,
between curves
1.9 and
3.4 MHz,
In the limitequally
kk → 0,spaced
the dispersion
remain
simi2
are
plotted.
The corresponding
lar to
those plotted
in Figure 4 at upper
large χhybrid
(shortfrequencies,
transverse
ω
+ symbols
on the ordinate.
Three
uh /2π, are indicated
wavelengths).
At long by
transverse
wavelengths,
however,
the
curves corresponding
to upper
conditions
with
upper hybrid
are cutoff at the
hybrid
frequency
ωuh . freAll
quencies
(above)band
the second
electron
gyroharmonic
curves in below
the frequency
below 2Ω
monotone decrease
frequency
are plotted
using
dashed black
(solidwavenumber,
red) lines.
ically in frequency
with
increasing
transverse
whereas all curves above 2Ωe increase in frequency from a
low-frequency cutoff, attain a maximum frequency at a secAnn. Geophys., 27, 2711–2720, 2009
D. L. Hysell and E. Nossa: FAI generation near 2e
: FAI GENERATION NEAR 2Ωe
In the limit kk → 0, the dispersion curves remain simiond, high-frequency cutoff, and then decrease
in frequency
lar to those plotted in Fig. 4 at large χ 2 (short transverse
once again. Mjølhus (1993) terms the branches to the left and
wavelengths). At long transverse wavelengths, however, the
right of the second cutoff upper hybrid and electron Bernstein
curves are cutoff at the upper hybrid frequency ωuh . All
modes, respectively.
curves in the frequency band below 2e decrease monotonWave trapping can be investigated using plots like Figure 4
ically in frequency with increasing transverse wavenumber,
through eikonal analysis (e.g. Landau and Lifshitz (1987);
whereas all curves above 2e increase in frequency from a
Weinberg (1962)), noting that wave
packets will propagate in
low-frequency cutoff, attain a maximum frequency at a secan inhomogeneous medium in such a way that frequency reond, high-frequency cutoff, and then decrease in frequency
mains invariant provided that the medium itself is not changonce again. Mjølhus (1993) terms the branches to the left and
ing rapidly. A wave packet launched at the low-frequency
right of the second cutoff upper hybrid and electron Bernstein
cutoff of an upper hybrid mode travels along a horizontal line
modes, respectively.
in Figure 4 as it penetrates into a depleted caviton. PropagatWave trapping can be investigated using plots like Fig. 4
ing into regions of lower density, it will ultimately encounter
through eikonal analysis (e.g. Landau and Lifshitz, 1987;
a turning point at the trough and be reflected, all the time reWeinberg, 1962), noting that wave packets will propagate in
maining on viable dispersion curves. A measure of the maxan inhomogeneous medium in such a way that frequency reimum depth of the caviton that can be formed through the
mains invariant provided that the medium itself is not changresonance instability is the difference in the two cutoff freing rapidly. A wave packet launched at the low-frequency
quencies of a given dispersion curve, which is related to the
cutoff of an upper hybrid mode travels along a horizontal line
maximum difference in plasma number densities at the cenas it penetrates
into caviton
a depleted
caviton. to
Propagating
in Fig.
ter
and 4walls
of the deepest
accessible
the wave
into
regions
of
lower
density,
it
will
ultimately
packet. Above 2Ωe , this difference decreases asencounter
the uppera
turningfrequency
point at the
trough and
be reflected,
all the time
rehybrid
approaches
an electron
gyroharmonic
from
maining
on
viable
dispersion
curves.
A
measure
of
the
maxbelow (Mjølhus, 1993). Since there are no second cutoffs
imum
depth
of frequency
the cavitonband
thatbelow
can be2Ωformed
through the
in
curves
in the
e , trapping cannot
resonance
instability
is
the
difference
in
the
two
cutoffcutfreoccur, i.e., wave packets launched at the low-frequency
quencies
of
a
given
dispersion
curve,
which
is
related
to
the
off cannot enter depleted cavitons and remain simultaneously
maximum
difference
in
plasma
number
densities
at
the
cenon a viable dispersion curve and a surface of constant freter and walls
of the would
deepest
accessible
to therather
wave
quency.
Such waves
becaviton
excluded
from cavitons
packet.
Above
2
,
this
difference
decreases
as
the
upper
e
than trapped within them in the kk → 0 limit.
hybrid
frequencyof
approaches
an electron
gyroharmonic
The inclusion
finite parallel
wavenumbers
in the from
calbelow
(Mjølhus,
1993).
Since
there
are
no
second
cutoffs
culation changes the picture described above by introducin curves
in the frequency
band below
2ein
, trapping
cannot
ing
or modifying
the low-frequency
cutoff
the dispersion
occur,
i.e.,
wave
packets
launched
at
the
low-frequency
curves at small transverse wavenumbers. The shapes of cutthe
off cannot
enter
depleted
cavitons
curves
in the
small
k⊥ limit
dependand
onremain
the sizesimultaneously
of kk2 as well
on on
a viable
dispersion
curve and
a surfacetheoflong
constant
freas
the collision
frequency.
However,
perpenquency.wavelength
Such wavescutoff
wouldisbe
excluded fromneither
cavitons
rather
dicular
a consequence
of collithan trapped
within them
in the kand
0 limit.mainly reflects
k →instead
sional
nor collisionless
damping
inclusion
of finite
infield
the caltheThe
reduced
restoring
force parallel
imposedwavenumbers
by the magnetic
for
culation
changes the
picturetodescribed
abovecurves
by introducwaves
propagating
obliquely
B. Dispersion
othering orapproaching
modifying the
cutoff in at
thelong
dispersion
wise
the low-frequency
upper hybrid frequency
transcurveswavelengths
at small transverse
wavenumbers.
of all
the
verse
dip sharply
in frequencyThe
as ashapes
result in
2 as well
curves
in
the
small
k
limit
depend
on
the
size
of
k
⊥
gyroharmonic frequency
bands. From the point of kview of
as on the
collision
frequency.
However,does
the not
long
perpeneikonal
analysis,
wave
trapping therefore
appear
to
dicular
wavelength
cutoff
is
a
consequence
neither
of
be prohibited at frequencies below the second electroncolligysional nor collisionless
damping
instead
mainly reflects
roharmonic
frequency when
finiteand
parallel
wavenumbers
are
the
reduced
restoring
force
imposed
by
the
magnetic
field for
permitted. As the high-frequency cutoff occurs at relatively
wavesvalues
propagating
othersmall
of k⊥ obliquely
for wavestoinB.
theDispersion
frequency curves
band below
wise
approaching
the
upper
hybrid
frequency
at
long
2Ωe and for those just below 2Ωe in particular, trappingtransmay
verse
wavelengths
sharply
in frequency
as a resultwavein all
be
limited
to wavesdip
with
relatively
small transverse
gyroharmonic
From value
the point
of view
of
lengths
in this frequency
band, withbands.
the limiting
related
to the
eikonal
analysis,
wave
trapping
therefore
does
not
appear
to
size of kk .
beAdditional
prohibitedinsights
at frequencies
below
the
second
electron
gycan be obtained by considering a simroharmonic
frequency
parallelrelation.
wavenumbers
are
plified,
algebraic
form when
of thefinite
dispersion
We will
permitted.
As
the
high-frequency
cutoff
occurs
at
relatively
regard kk as being sufficiently small to permit the approxismall values
of kω⊥ −fornΩwaves
mation
that, with
anthe
→frequency
∞, and Wband
(an ) below
→ 0.
e 6= 0,in
www.ann-geophys.net/27/2711/2009/
D. L. Hysell and E. Nossa: FAI generation near 2e
2717
2e and for those just below 2e in particular, trapping may
be limited to waves with relatively small transverse wavelengths in this band, with the limiting value related to the
size of kk .
Additional insights can be obtained by considering a simplified, algebraic form of the dispersion relation. We will
regard kk as being sufficiently small to permit the approximation that, with ω − ne 6 = 0, an → ∞, and W (an ) → 0.
This limit excludes Landau (cyclotron) damping and reduces
the dispersion relation in Eq. (1) to:
(
)
P∞
ω+iν
2
kd2 1 − n=−∞ ω+iν−ne 3n (χ )
(ω,k) = 1 + 2
(2)
P
iν
2
k 1− ∞
n=−∞ ω+iν−n 3n (χ )
e
The sums in this formula can be further simplified with the
following identities:
In (z) = I−n (z)
∞
X
1=
3n (z)
n=−∞
with the result
(3)
(ω,k)
=
2
k 2 e−χ
1 − 2ωp2 ⊥2 2
k χ
∞
X
n2 I
(χ 2 )
ω + iν
n
ω n=1 (ω + iν)2 − n2 2e
where only the n = 0 term in the denominator of Eq. (2) has
been retained. We next consider an expansion of Eq. (3)
in orders of the χ 2 parameter, focusing on behavior at long
transverse wavelengths and making use of the following expansion for In :
In (z) =
1
z
2
n X
∞
k=0
1 2
4z
k
k!0(n + k + 1)
The zero and first order terms in the dispersion relation can
then be shown to be:
k 2 X(1 + iZ)
O(χ ) : 1 − ⊥2
k (1 + iZ)2 − Y 2
k2
3X(1 + iZ)Y 2
O(χ 2 ) : −χ 2 ⊥2
2
k ((1 + iZ) − Y 2 )((1 + iZ)2 − 4Y 2 )
0
where we make use of the standard notation X ≡ ωp2 /ω2 ,
Y ≡ e /ω, and Z ≡ ν/ω and consider only the n = 1 and
n = 2 terms in the sum in Eq. (3). Note that investigating
asymmetric collisionless damping above and below gyroharmonic frequencies necessitates retaining higher order terms
in χ 2 as well as larger values of n in the sum. We truncate here to maintain consistency with other work, focusing
on the effects of finite parallel wavenumbers and relegating
asymmetry to future analysis.
www.ann-geophys.net/27/2711/2009/
Combining the components above yields the following,
fluid-theory approximation for the dielectric permittivity:
(4)
(ω,k)
= 1−
2
k⊥
k2
1+Y2
3χ 2 Y 2
X
1 + iZ
+
1−Y2
1 − Y 2 1 − 4Y 2
!
In calculating Eq. (4), we regard the second and third terms
in the parentheses as small corrections associated with collisional and thermal effects, respectively, and neglect additional cross terms combining both effects. The = 0 result
is then consistent with the dispersion relations analyzed by
Mjølhus (1993) and Kosch et al. (2007), for example, in the
appropriate limiting regimes.
Dysthe et al. (1982) and then Mjølhus (1993) used dielectric permittivity functions similar to Eq. (4) to develop ordinary differential equations describing upper hybrid wave
trapping in depleted cavitons in one dimension transverse to
the geomagnetic field. They did this by linearizing Gauss’
law for dielectric media, ∇ · (E) = 0, where the electric
field includes the electromagnetic pump field and the electrostatic response. The linearization introduced an additional
small correction associated with the density perturbation of
the caviton. Equation (4) adds a fourth correction due to finite parallel wavenumbers. Following the formalism of the
earlier works and including the finite kk correction, which is
formulated somewhat differently here than by Dysthe et al.
(1982), we can derive the following coupled system of differential equations describing the transverse electric field of
the upper hybrid wave E1 and an auxiliary field F1 :
F1 + λ2
d 2 E1
+ (0 − iδ − N (x))E1 = pE◦ N (x)
dx 2
!
−d 2
2
+ kk F1 = kk2 E1
dx 2
(5)
(6)
where x is the direction transverse to the geomagnetic field,
E◦ represents the pump mode field amplitude, and where the
other symbols are defined by
λ2 = 3ρ 2 Y 2 /(1 − 4Y 2 )
0 = 1 − X/(1 − Y 2 )
δ = Z(1 + Y 2 )/(1 − Y 2 )
N = δn/n◦
pE◦ N (x) = −k̂ · δ · E◦
with p a geometric factor.
Figure 5 shows representative solutions of Eqs. (5) and
(6) (with E1 (x) plotted here) demonstrating wave trapping.
The top panel represents 0 − N (x) for a depleted caviton
and illustrates how the dielectric constant is positive (negative) inside (outside) the depletion. Neglecting the terms
involving kk2 , Eqs. (5) and (6) represent a second-order differential equation, which is expected to exhibit oscillating
Ann. Geophys., 27, 2711–2720, 2009
2718
8
Fig.5.5. Representative
solutions of
equations
(5)Nand
Top panel:
Fig.
Top panel: Functional
form
of 0 −
(x)(6).
corresponding
Functional
form
of
ǫ
−
N
(x)
corresponding
to
a
depleted
caviton.
0
to a depleted caviton. Second panel: Solution
for the case
kk2 = 0
Second
panel: Solution for the case kk2 = 0 and2λ2 > 0. Third
2
and λ > 0. Third panel: Solution
for the case k = 0 and λ2 < 0
panel: Solution for the case kk2 = 0 and λ2 < k0 Bottom panel:
Bottom
Solution
caseλk2k2 >
λ2 < 0.boundary
Dirichlet
Solutionpanel:
for the
case kk2for>the
0 and
< 00.and
Dirichlet
boundary
conditions
are
applied
at
either
end
of
the
conditions are applied at either end of the abscissa. abscissa.
andThe
evanescent
behavior
inside
and outside
the depleted
rethird panel
of Figure
5 shows
a representative
solugion,
λ2 > 0applies
and δ 1. This
tion respectively,
for the λ2 < provided
0 case, which
when
ω <implies
2Ωe .
wave
insideisthe
caviton,
as illustrated
by the
second
Here,trapping
the situation
reversed,
such
that the wave
is evanescent of
within
therefore
excluded
from the
caviton. within
The
panel
Fig. and
5. The
number
of waveforms
contained
caviton
will experience
deficit
heating to
compared
the
depletion
depends ona the
sizeofofwave
λ2 relative
the size
this
case, andcontrolling
the resonance
oftoits
N (x). Wheninthe
parameter
0 −instability
N (x) ap0 −surroundings
will not an
occur.
Note that
trapped
solutions can
reemerge
if
proaches
eigenvalue
of the
homogeneous
system
of equalarge the
values
of δ are
considered,
dissipation
associ-of
tions,
solutions
grow
without although
bound. This
is the nature
ated
with collisional
damping
tend
to be stabilizing.
the
resonance
instability,
with will
actual
resonances
correspondthe bottom
of Figure 5 fitting
shows precisely
a representaing Finally,
to integral
numberspanel
of waveforms
into
2
tivedepleted
solutionzone.
to (5)Various
and (6)limiting
for the case
<
but withhave
the
the
casesλof
the0problem
kk2 terms
restored
this time.
The resulting,
fourth-order
difet al. (1982)
and Mjølhus
been
described
in detail
by Dysthe
ferential
equation
is
seen
to
be
capable
of
producing
trapped
(1993).
solutions
in panel
the ω <
The morphology
of thesolusoe regime.
The third
of2Ω
Figure
5 shows
a representative
lution, number
of waveforms in the depletion, etc., are con2
tion for the λ < 0 case, which
applies when ω < 2e . Here,
trolled by the ratios of k 2 , λ2 , and ǫ − N (x) in a more
the situation is reversed, ksuch that the0 wave is evanescent
within and therefore excluded from the caviton. The caviton
Ann. Geophys., 27, 2711–2720, 2009
D. L. Hysell and E. Nossa: FAI generation near 2e
: FAI GENERATION NEAR 2Ωe
will experience a deficit of wave heating compared to its surcomplicated way than was the second-order system. The
roundings in this case, and the resonance instability will not
boundary between oscillating and evanescent wave behavior
occur. Note that trapped solutions can reemerge if large valdepends not only on the sign of ǫ but also on its curvature,
ues of δ are considered, although dissipation associated with
for example.
collisional damping will tend to be stabilizing.
Finally, the bottom panel of Fig. 5 shows a representative
solution
to Eqs. (5) and (6) for the case λ2 < 0 but with the
4 Discussion
2
kk terms restored this time. The resulting, fourth-order differential equation
is seen to(FoE)
be capable
trapped
Sustained
critical frequency
valuesofofproducing
about 3 MHz
solutions
in the ω < 2
morphology
of the
during
geomagnetically
quiet
summer The
afternoons
afford the
e regime.
solution, of
number
in the depletion,
possibility
routineofE waveforms
region ionospheric
modificationetc.,
ex- are
2 , and
periments
over
to operate
controlled
by HAARP,
the ratioswhich
of kk2is, λable
0 − N at
(x)frequenin a more
cies
as low as way
2.75 than
MHz was
in practice.
Absent geomagnetic
complicated
the second-order
system. The
activity,
the between
experiments
are relatively
immune towave
variations
boundary
oscillating
and evanescent
behavior
independs
heating and
propagation
thatalso
can interfere
with
not only
on the conditions
sign of but
on its curvature,
the
forconsistency
example. of F region experiments. Artificial E region FAIs can be monitored using the coherent scatter radar
near Homer, the result being a well controlled experiment.
Most
of the experiments performed so far were undertaken
4 Discussion
at heating frequencies below the second electron gyroharmonic.
Preliminary
argue(FoE)
that thevalues
thermal
Sustained
critical results
frequency
ofOTSI,
aboutwave
3 MHz
trapping,
and the resonance
instability
at work atafford
these the
during geomagnetically
quiet
summerareafternoons
experiments
et al.,
2009). ionospheric modification expossibility (Nossa
of routine
E region
The radarover
experiments
differentiate
periments
HAARP,described
which is here
able do
to not
operate
at frequenclearly
phenomena
taking
placegeomagnetic
at pump
cies asbetween
low as heating
2.75 MHz
in practice.
Absent
frequencies
and above
secondimmune
electron to
gyroharactivity, thebelow
experiments
are the
relatively
variations
monic
frequency
in
the
main.
Upper
hybrid
wave
trappingwith
in heating and propagation conditions that can interfere
in density cavitons therefore does not appear to be ruled out
the consistency of F region experiments. Artificial E reat pump frequencies below 2Ωe . Analysis suggests that a
gion FAIs can be monitored using the coherent scatter radar
second, low-frequency cutoff in the upper hybrid wave disnear Homer, the result being a well controlled experiment.
persion relation, which is required for wave trapping, can be
Most of the experiments performed so far were undertaken
introduced at long perpendicular wavelengths by including
at heating frequencies below the second electron gyroharthe effects of finite parallel wavenumbers. Below 2Ωe , the
monic. Preliminary
that the thermal
OTSI, wave
high-frequency
cutoff results
occurs argue
at a relatively
small transverse
trapping,
and
the
resonance
instability
are
at
work
at these
wavenumber that depends on the value of kk . This suggests
et
al.,
2009).
experiments
(Nossa
that the trapped waves may not be able to attain very short
The radar
experiments
described
here
not differentiate
transverse
wavelengths
below
2Ωe , and
thatdosome
heatingclearly
between
heating
phenomena
taking
pump
related phenomena may therefore be interrupted, place
even ifat5-m
frequencies
below
and
above
the
second
electron
gyroharscale FAI generation is not.
monic
frequency
in the including
main. Upper
wave et
trapping
A number
of studies,
thosehybrid
by Dysthe
al.
in
density
cavitons
therefore
does
not
appear
to
be
ruled
(1982) and Mjølhus (1993), considered finite parallel out
at pump frequencies
below 2
suggests
that a
wavenumber
effects, focusing
mainly
on Landau
damping,
e . Analysis
second,
low-frequency
cutoff in
upper
hybrid
waveindisand
the associated
suppression
of the
upper
hybrid
waves
the
immediate
vicinity
of electron
gyroharmonic
frequencies.
persion
relation,
which
is required
for wave trapping,
can be
The
conditionatforlong
neglecting
Landau damping
at allby
χ2 including
is that
introduced
perpendicular
wavelengths
ωthe
− nΩ
|kkfinite
|vte . parallel
(A less restrictive
condition
was2
also
effects
wavenumbers.
Below
e ≫ of
e , the
2
derived
for the limit
of small
.)a In
view ofsmall
the fact
that
high-frequency
cutoff
occursχat
relatively
transverse
we
have not as athat
ruledepends
observedonwave
suppression
∼50
wavenumber
the value
of kk . within
This suggests
kHz
thetrapped
second electron
gyroharmonic
frequency,
thatofthe
waves may
not be able
to attain this
veryimshort
−1
plies
an upper
bound of about
= e4, m
in thesome
E region,
transverse
wavelengths
belowkk2
and that
heatingwhich
is phenomena
an order of may
magnitude
larger
than the value
related
therefore
be interrupted,
evenused
if 5-m
toscale
compute
Figure
4
(where
the
effects
of
k
were
accentuk
FAI generation is not.
atedAfornumber
plottingofpurposes)
40 times
the by
value
used to
Dysthe
et al.
studies, and
including
those
compute the bottom panel of Figure 5. Evaluating experi(1982) and Mjølhus (1993), considered finite parallel
mental and theoretical evidence from a number of sources,
wavenumber effects, focusing mainly on Landau damping,
and the associated suppression of upper hybrid waves in
www.ann-geophys.net/27/2711/2009/
D. L. Hysell and E. Nossa: FAI generation near 2e
the immediate vicinity of electron gyroharmonic frequencies. The condition for neglecting Landau damping at all
χ 2 is that ω − ne |kk |vte . (A less restrictive condition
was also derived for the limit of small χ 2 .) In view of the
fact that we have not as a rule observed wave suppression
within ∼50 kHz of the second electron gyroharmonic frequency, this implies an upper bound of about kk =4 m−1 in the
E region, which is an order of magnitude larger than the value
used to compute Figure 4 (where the effects of kk were accentuated for plotting purposes) and 40 times the value used
to compute the bottom panel of Fig. 5. Evaluating experimental and theoretical evidence from a number of sources,
Mjølhus (1993) found upper limits on kk of order unity in
MKS units in the F region. A reasonable expectation in practice might be for kk to match approximately the wavenumber of the pump wave. In the future, it may be possible to
estimate the parallel wavenumber from observations of FAI
suppression at pump frequencies in the stop band closer to
the electron gyroharmonic. Measuring a frequency crossing
in the E region is somewhat more difficult than it is in the F
region given the relatively small altitude excursion of the E
layer during the daytime (see Kosch et al., 2007, for reference). Sporadic E layers may pose a more suitable target for
this kind of experiment, although the disturbed conditions
that often accompany them at high latitudes undermine the
controlled nature of the experiment.
Radar interferometry is routinely used to measure the aspect sensitivity of FAIs in the ionosphere (e.g. Farley et al.,
1981). The results from such experiments generally refer to
the RMS deviation or spreading of the scattering wavevector about the normal direction, which is easy to measure,
rather than a systematic bias in a given direction, which is
not. We might equally well interpret the parallel wavenumber in question here as an RMS deviation about the normal
direction. The mathematics is essentially identical, although
the physical picture is different. However, eikonal analysis
in the vertical (parallel to B) direction also suggests that the
upper hybrid waves should acquire a bias toward finite kk
while propagating, even if launched transverse to B, suggesting that a combination of bias and spread in kk could
be present. Note that, as was pointed out by Mjølhus (1993),
the parallel wavenumber of the upper hybrid wave is not the
same thing as the parallel wavenumber of the striations visible to radar, and so radar interferometry cannot by itself resolve the role or interpretation of parallel wavenumbers in
this context. Note also that it is the aspect sensitivity of the
striations that concerned Grach (1979) in his calculation of
the threshold for thermal parametric instability.
Although the data quality is marginal, our sole example
of artificial FAIs generated in a sporadic E layer suggests
that wave trapping at pump frequencies below 2e can be
suppressed under the right circumstances. The ionogram for
the event in question placed the sporadic E layer at about
110 km altitude, close to the critical height in the daytime.
We therefore cannot conclude that either pump frequency
www.ann-geophys.net/27/2711/2009/
2719
was closer to 2e than in the daytime experiments. Given
that the upper-hybrid interaction height in a sharply-peaked
sporadic E layer is apt to be extraordinarily narrow, we may
speculate that parallel wavenumbers may be drastically limited by the requirement that the trapped waves may not propagate vertically out of the layer. This imposes an additional
limit on the shortest transverse wavelength for which trapping can occur. It could be that wave trapping at pump frequencies below 2e either cannot occur in a narrow sporadic
E layer or cannot excite cavitons sufficiently small to appear
as strong targets to a VHF radar. Further experiments involving additional diagnostics are needed to elucidate this
behavior.
Acknowledgements. The authors are grateful for help received from
the NOAA Kasitsna Bay Laboratory, its Director Kris Holderied,
Lab Manager Mike Geagel, and Lab Director Connie Geagel. We
also appreciate helpful comments from C. La Hoz from the University of Tromsø who is presently on leave at Cornell. This work was
supported by the High Frequency Active Auroral Research Program
(HAARP) and by the Office of Naval Research and the Air Force
Research Laboratory under grant N00014-07-1-1079 to Cornell.
Topical Editor M. Pinnock thanks A. Streltsov and B. Isham
for their help in evaluating this paper.
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